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All questions of Data Sufficiency: 700 Level for GMAT Exam

a, b, and c are three distinct positive integers. What is the product abc?

(1) a + b + c = 7
(2) ab + bc + ca = 14

a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient

d)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed

e)
EACH statement ALONE is sufficient to answer the question asked

Correct answer is option 'E'. Can you explain this answer?

Bhavya Khanna answered

Statement Analysis:

Statement 1:
- a + b + c = 7

Statement 2:
- ab + bc + ca = 14

Combined Analysis:
- Both statements alone do not provide enough information to determine the product abc.
- While statement 1 gives the sum of the three integers, statement 2 gives the sum of their products.
- To find the product abc, we need to know the individual values of a, b, and c.
Therefore, option 'E' is the correct answer as statements (1) and (2) together are not sufficient to answer the question asked.

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‎[x] denotes to be the least integer no less than x. Is [2d] = 0?
(1) [d] = 0
(2) [3d] = 0
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Niharika Sen answered

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What is the percentage of alcohol in solution X?
(1) If 50 liters alcohol is added, X will contain 60% alcohol.
(2) If the volume of water, equivalent to that of the total solution, is added, X will contain 20% alcohol.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Hridoy Gupta answered

Statement Analysis:
Given: We need to find the percentage of alcohol in solution X.

Statement 1:
- Adding 50 liters of alcohol will result in solution X containing 60% alcohol.
- This statement alone is not sufficient to determine the percentage of alcohol in solution X without additional information.

Statement 2:
- Adding the volume of water equivalent to the total solution will result in solution X containing 20% alcohol.
- This statement alone is sufficient to determine the percentage of alcohol in solution X, as it provides a clear indication of the alcohol content after dilution.

Conclusion:
- Statement 2 alone is sufficient to answer the question asked.
- Therefore, the correct answer is option 'B' which states that Statement 2 alone is sufficient, but Statement 1 alone is not sufficient to answer the question asked.

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In a class of 100 students, how many are boys? (There is at least 1 girl)
(1) In IIT - entrance exam, 10% of girls in the class failed to qualify.
(2) There are more than 80 boys in the class.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Palak Saha answered

Statement 1: In IIT - entrance exam, 10% of girls in the class failed to qualify.

Statement 2: There are more than 80 boys in the class.

To find the number of boys in the class, we need to consider both statements together.

Analysis of Statement 1:
From statement 1, we know that 10% of girls failed to qualify in the IIT-entrance exam. However, we do not have any information about the total number of girls in the class or the total number of students who failed to qualify. Therefore, statement 1 alone is not sufficient to answer the question.

Analysis of Statement 2:
Statement 2 tells us that there are more than 80 boys in the class. However, we do not have any information about the total number of students or the number of girls in the class. Therefore, statement 2 alone is not sufficient to answer the question.

Analysis of Statements 1 and 2 together:
Combining both statements, we can infer the following:

- Since there is at least 1 girl in the class, the remaining students must be boys.
- From statement 2, we know that there are more than 80 boys in the class.

Therefore, the number of boys in the class is greater than 80.

Since both statements together provide enough information to determine that the number of boys in the class is greater than 80, the correct answer is option C, i.e., BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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If x and y are positive integers greater than 1, is the unit digit of x * y greater than 5?
(A) x is a factor of 85.
(B) The units digit of y² and y⁷is the same.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Devansh Shah answered

Understanding the Problem
The objective is to determine whether the unit digit of the product x * y is greater than 5, given that x and y are positive integers greater than 1.
Analysis of Statement (1)
- Statement (1): x is a factor of 85.
- Factors of 85 are 1, 5, 17, and 85. Since x must be greater than 1, possible values for x are 5 and 17.
- Now, evaluate the unit digits:
- If x = 5: The unit digit of x * y will depend on the unit digit of y (which can be any digit).
- If x = 17: The unit digit of x * y will also depend on the unit digit of y.
- Conclusion: This statement alone does not determine if the unit digit of x * y is greater than 5.
Analysis of Statement (2)
- Statement (2): The units digit of y^2 and y^7 is the same.
- The units digit of y^2 can only match y^7 if the units digit of y is 0, 1, 5, or 6.
- Evaluating these:
- y ending in 0: y^2 = 0, y^7 = 0 (not > 5).
- y ending in 1: y^2 = 1, y^7 = 1 (not > 5).
- y ending in 5: y^2 = 5, y^7 = 5 (not > 5).
- y ending in 6: y^2 = 6, y^7 = 6 (could yield a product > 5 depending on x).
- Conclusion: This statement alone does not provide a definitive answer either.
Combining Statements (1) and (2)
- Even when combined, the values of x (5 or 17) and y (0, 1, 5, or 6) do not conclusively determine if the unit digit of x * y is greater than 5. Each combination still holds possibilities that can lead to unit digits less than or equal to 5.
Final Conclusion
- Correct Answer: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. Hence, the answer is option 'E'.

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On Christmas, a class teacher gave a gift pack to each student present in a class. Each pack had x chocolates, y biscuits and z toffees. How many students were there in the class?
(1) Each student received chocolates, biscuits and toffees in the ratio 1:2:4.
(2) The class teacher distributed a total of 7 chocolates, 14 biscuits and 28 toffees and the number of students present in the class was more than 1.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Pioneer Academy answered

Statement (1): Each student received chocolates, biscuits, and toffees in the ratio 1:2:4.
From this statement, we know that the ratio of chocolates to biscuits to toffees is 1:2:4. However, we don't have any specific values for x, y, or z. There are multiple possible combinations that satisfy this ratio, such as x = 1, y = 2, and z = 4 or x = 2, y = 4, and z = 8. Without more information, we cannot determine the exact values of x, y, and z or the number of students in the class. Statement (1) alone is not sufficient.

Statement (2): The class teacher distributed a total of 7 chocolates, 14 biscuits, and 28 toffees, and the number of students present in the class was more than 1.
From this statement, we have specific values for the total number of chocolates, biscuits, and toffees distributed. However, we still don't have enough information to determine the exact values of x, y, and z or the number of students in the class. The given values could be divided among different numbers of students that satisfy the conditions. For example, the values could be divided among 2 students, where each student receives 3 chocolates, 7 biscuits, and 14 toffees. Alternatively, they could be divided among 7 students, where each student receives 1 chocolate, 2 biscuits, and 4 toffees. Statement (2) alone is not sufficient.

Since neither statement alone is sufficient to answer the question, the correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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A contractor combined x tons of a gravel mixture that contained 10 percent gravel G, by weight, with y tons of a mixture that contained 2 percent gravel G, by weight, to produce z tons of a mixture that was 5 percent gravel G, by weight. What is the value of x ?
(1) y = 10
(2) z = 16
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Amar Chakraborty answered

To find the value of x, we need to determine the amount of gravel G in the mixture produced.

Let's break down the problem step by step:

1. The contractor combined x tons of a gravel mixture that contained 10 percent gravel G, by weight.
2. The contractor also combined y tons of a mixture that contained 2 percent gravel G, by weight.
3. The resulting mixture produced z tons and contained 5 percent gravel G, by weight.

From this information, we can set up the following equations:

Equation 1: (0.10x + 0.02y) / z = 0.05

Statement 1: y = 10
This statement gives us the value of y, but it does not provide any information about x or z. Therefore, it is not sufficient to find the value of x.

Statement 2: z = 16
This statement gives us the value of z, but it does not provide any information about x or y. Therefore, it is not sufficient to find the value of x.

Statements 1 and 2 together:
With both statements, we have y = 10 and z = 16. Substituting these values into Equation 1, we get:

(0.10x + 0.02(10)) / 16 = 0.05

Simplifying the equation, we have:

(0.10x + 0.2) / 16 = 0.05

Multiplying both sides by 16, we get:

0.10x + 0.2 = 0.8

Subtracting 0.2 from both sides, we have:

0.10x = 0.6

Dividing both sides by 0.10, we get:

x = 6

Therefore, with both statements together, we can determine the value of x. Hence, the correct answer is option (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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For any integers x and y, min(x, y) and max(x, y) denote the minimum and the maximum of x and y, respectively. For example, min (5, 2) = 2 and max(5, 2) = 5. For the integer w, what is the value of min(10, w) ?
(1) w = max(20, z) for some integer z.
(2) w = max(10, w)
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

EduRev GMAT answered

Statement 1: w = max(20, z) for some integer z.
This statement tells us that the value of w is equal to the maximum of 20 and some integer z. However, it doesn't provide any information about the specific value of z or how it relates to the value of w. Therefore, statement 1 alone is not sufficient to determine the value of min(10, w).

Statement 2: w = max(10, w)
This statement tells us that the value of w is equal to the maximum of 10 and w itself. This implies that w must be greater than or equal to 10. However, it doesn't provide any specific value for w. Therefore, statement 2 alone is not sufficient to determine the value of min(10, w).

When we consider both statements together, we know that w is equal to the maximum of 10 and itself (w = max(10, w)). This means that w must be greater than or equal to 10. Additionally, statement 1 tells us that w is equal to the maximum of 20 and some integer z. Since 20 is greater than 10, we can conclude that w must be equal to 20.

Therefore, each statement alone is sufficient to determine the value of min(10, w), and the answer is option D: EACH statement ALONE is sufficient to answer the question asked.

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There are 3 different positive integers. If their average (the arithmetic mean) is 8, what are their values?
1) The largest integer is twice the smallest integer.
2) One of them is 9.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

EduRev GMAT answered

Statement 1: The largest integer is twice the smallest integer.
This statement gives us a relationship between the largest and smallest integers, but it doesn't provide specific values. Without knowing the actual values of the integers, we cannot determine their individual values or the value of the middle integer. Therefore, statement 1 alone is not sufficient to determine the values of the integers.

Statement 2: One of them is 9.
This statement gives us the specific value of one of the integers, which is 9. However, it doesn't provide any information about the other two integers or their relationship. Therefore, statement 2 alone is not sufficient to determine the values of the integers.

When we consider both statements together, we can conclude the following:

The largest integer is twice the smallest integer.
One of the integers is 9.

From statement 2, we know that one of the integers is 9. If we consider this as the largest integer, then the smallest integer would be 9/2 = 4.5, which is not a positive integer. Therefore, the largest integer cannot be 9.

Since the integers are positive and the largest integer is twice the smallest integer, we can conclude that the smallest integer is 1, and the largest integer is 2. The middle integer is then (1 + 2 + 9)/3 = 4.

Therefore, the values of the integers are 1, 4, and 9.

The answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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For any integers x and y, min(x,y) and max(x,y) denote the minimum and the maximum of x and y, respectively. For example, min(5,2) = 2 and max(5,2) = 5. For the integers a and b, what is the value of max(a,b)?
(1) a > b
(2) ab = −1
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Geetika Sarkar answered

Understanding the Problem
To find the value of max(a, b), we need to analyze the information given in the statements.
Evaluating Statement (1)
- Statement (1): a > b
- If a is greater than b, then we can directly conclude that max(a, b) = a.
- Thus, this statement alone is sufficient to determine the value of max(a, b).
Evaluating Statement (2)
- Statement (2): ab = -1
- This implies that one of the integers is positive and the other is negative (since their product is negative).
- However, without knowing the specific values of a and b, we cannot definitively say which one is larger. For example:
- If a = 1 and b = -1, then max(a, b) = 1.
- If a = -1 and b = 1, then max(a, b) = 1.
- If a = -1 and b = 1, then max(a, b) = 1.
- Therefore, this statement alone is not sufficient to find max(a, b).
Conclusion
- Final Analysis
- Statement (1) is sufficient.
- Statement (2) is not sufficient.
Thus, the correct answer is option B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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If x is a positive integer, what is the units digit of x?
(1) The units digit of x/10 = 4
(2) The tens digit of 10x = 5
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Prateek Gupta answered

To find the units digit of a positive integer x, we need to analyze the given information in both statements.

Statement (1): The units digit of x/10 = 4
This statement tells us that the remainder when x is divided by 10 is 4. In other words, x ends with a 4 in the units place. However, we do not have enough information to determine the exact value of x or its units digit. For example, x could be 4, 14, 24, 34, etc. In each case, the units digit is 4, but the exact value of x varies. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): The tens digit of 10x = 5
This statement tells us that the tens digit of 10x is 5. Since 10x is a multiple of 10, the units digit of 10x is always 0. Therefore, the units digit of x must also be 0. For example, x could be 5, 15, 25, 35, etc. In each case, the units digit is 0. Thus, statement (2) alone is sufficient to answer the question.

Combining both statements, we know that the units digit of x is 0 and that it also ends with a 4. The only number that satisfies both conditions is 4. Therefore, both statements together are not sufficient to answer the question.

Therefore, the correct answer is option B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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If x is a member of the set {0.3, 0.22, 0.265}, what is the value of x?
(1) 0.25 < x < 0.35
(2) 0.2 < x < 0.28
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement 1: 0.25 < x < 0.35
This statement tells us that the value of x lies between 0.25 and 0.35. Comparing this range to the set {0.3, 0.22, 0.265}, we can see that x = 0.3 satisfies this condition. However, we cannot determine if any of the other values in the set satisfy this condition.

Statement 2: 0.2 < x < 0.28
This statement tells us that the value of x lies between 0.2 and 0.28. None of the values in the set {0.3, 0.22, 0.265} satisfy this condition.

Considering both statements together, we can conclude that the only value in the set that satisfies both conditions is x = 0.3. Therefore, both statements together are sufficient to answer the question. The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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The ratio of water to alcohol in a 14 cup container is 2:5. Determine the new volume of the liquid in the container.
1. Water is increased by 14%.
2. Mixture whose ratio of water to alcohol is 4:5 is added to that in the container.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

EduRev GMAT answered

Statement (1): Water is increased by 14%.

If the water in the container is increased by 14%, we can calculate the new volume of water. However, this statement alone does not provide any information about the alcohol or the total volume of the mixture. Therefore, we cannot determine the new volume of the liquid in the container based solely on this statement.

Statement (2): A mixture whose ratio of water to alcohol is 4:5 is added to that in the container.

This statement provides information about a new mixture that is added to the container, but it doesn't specify the volume of the new mixture or how it relates to the original mixture. Without knowing the volume of the added mixture or how it combines with the existing mixture, we cannot determine the new volume of the liquid in the container based solely on this statement.

Since neither statement alone is sufficient to answer the question, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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A function "a#b" is defined for integers a and b such that a#b = a+b if both a and b are odd, and a#b = (a*b)/2 otherwise. For integers x and y, what is x#y?
(1) |y| < 1.
(2) x = 14k, where 'k' is a non negative integer.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): |y| < 1.
This statement implies that the absolute value of y is less than 1. However, it doesn't provide any specific information about the values of x and y or their oddness/evenness. Without knowing the values of x and y or their properties, we cannot determine the value of x#y. Statement (1) alone is not sufficient.

Statement (2): x = 14k, where 'k' is a non-negative integer.
This statement provides information about the value of x, stating that it is equal to 14k, where k is a non-negative integer. However, it doesn't provide any information about the value of y or its oddness/evenness. Without knowledge of the value of y or its properties, we cannot determine the value of x#y. Statement (2) alone is not sufficient.

Combining both statements:
By considering both statements together, we know that |y| < 1 and x = 14k, where 'k' is a non-negative integer. However, even with this combined information, we still don't have any insight into the specific values of x and y or their oddness/evenness. Therefore, we cannot determine the value of x#y.

As a result, the correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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A gasoline mixture contains g liters gasoline, b liters benzene, and a liters of other additives. If there are 12 liters of gasonline mixture in an automobile tank, how many liters of additives are in the tank?

(1) If 3 liters of additives were added to the mixture, benzene would only account for 30% of the total mixture.
(2) (b + g + 4.5)/3 = 3g/2
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): If 3 liters of additives were added to the mixture, benzene would only account for 30% of the total mixture.

This statement alone provides information about the benzene content relative to the additives when 3 liters of additives are added. However, it does not give us any specific information about the quantities of gasoline, benzene, or additives in the original mixture. Therefore, it is not sufficient to answer the question.

Statement (2): (b + g + 4.5)/3 = 3g/2

This equation provides a relationship between the quantities of gasoline (g) and benzene (b) in the mixture. However, it does not provide any information about the quantity of additives or the total quantity of the mixture. Without knowing the specific quantities of gasoline, benzene, and additives, we cannot determine the number of liters of additives in the tank. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have some information about the benzene content relative to additives (from statement 1) and a relationship between gasoline and benzene quantities (from statement 2). However, we still lack the necessary information to determine the actual quantities or solve for the number of liters of additives. Therefore, the answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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Six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries sent the same number of representatives. Of the six countries, if Country A sent the second greatest number of representatives, did Country A send at least 10 representatives?
(1) One of the six countries sent 41 representatives to the congress
(2) Country A sent fewer than 12 representatives to the congress
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Pioneer Academy answered

Statement 1: One of the six countries sent 41 representatives to the congress.
This statement tells us that one of the countries sent 41 representatives, but it doesn't provide any information about the number of representatives sent by Country A. Therefore, statement 1 alone is not sufficient to determine whether Country A sent at least 10 representatives.

Statement 2: Country A sent fewer than 12 representatives to the congress.
This statement provides information specifically about Country A, stating that it sent fewer than 12 representatives. However, it doesn't give us the exact number of representatives sent by Country A. Therefore, statement 2 alone is not sufficient to determine whether Country A sent at least 10 representatives.

When we consider both statements together, we still don't have enough information to determine whether Country A sent at least 10 representatives. Statement 1 tells us that one of the countries sent 41 representatives, but we don't know if that country is Country A or if Country A sent more or fewer representatives than that. Statement 2 tells us that Country A sent fewer than 12 representatives, but we don't know the exact number.

Without additional information about the number of representatives sent by Country A or the relationship between Country A and the country that sent 41 representatives, we cannot determine whether Country A sent at least 10 representatives.

Therefore, the answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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If S is a sequence of consecutive multiples of 3, how many multiples of 9 are there in S?
(1) There are 15 terms in S.
(2) The greatest term of S is 126.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Wizius Careers answered

To determine the number of multiples of 9 in the sequence S, let's analyze the information provided in each statement:

Statement (1): There are 15 terms in S.
This statement tells us the total number of terms in the sequence S. However, it does not provide any specific information about the values of the terms or their distribution. Since we don't know the exact values of the terms in S, we cannot determine the number of multiples of 9 in the sequence based solely on this statement. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): The greatest term of S is 126.
This statement gives us information about the largest term in the sequence S. However, it does not provide any information about the other terms or their distribution. Knowing the largest term alone is not enough to determine the number of multiples of 9 in the sequence. For example, S could include multiples of 3 that are not multiples of 9. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements:
By combining the information from both statements, we know that there are 15 terms in the sequence S and the largest term is 126. However, even with this combined information, we still don't have any direct information about the specific values of the terms or their distribution. For example, S could contain multiples of 3 that are not multiples of 9. Therefore, the statements together are not sufficient to answer the question.

Since neither statement alone nor the statements together provide enough information to answer the question, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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In what ratio did a grocer mix three varieties of tea that cost $6 per kg, $7 per kg and $8 per kg respectively, to make a profit of 10% by selling the mixture at $7.70 per kg?
(1) The grocer mixed 135 kg of the variety of tea that costs $7 per kg.
(2) The weight of the mixture was 315 kg.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

EduRev GMAT answered

Statement (1): The grocer mixed 135 kg of the variety of tea that costs $7 per kg.

This statement tells us the weight of one of the tea varieties used in the mixture. However, it doesn't provide any information about the weights or costs of the other two varieties or the total weight of the mixture. Therefore, statement (1) alone is not sufficient to determine the ratio in which the teas were mixed.

Statement (2): The weight of the mixture was 315 kg.

This statement provides information about the total weight of the mixture but doesn't provide any information about the weights or costs of the individual tea varieties used in the mixture. Therefore, statement (2) alone is not sufficient to determine the ratio in which the teas were mixed.

Now let's consider both statements together. From statement (1), we know that the weight of one of the tea varieties is 135 kg. From statement (2), we know that the total weight of the mixture is 315 kg.

However, even when considering both statements together, we still don't have any information about the costs of the tea varieties or the specific ratios in which they were mixed. Therefore, the information provided by both statements is not sufficient to determine the ratio in which the teas were mixed.

Hence, the answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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A sports team played 100 games last season. Did this team win at least half of the games it played last season?
(1) The team won 60% of its first 65 games last season.
(2) The team won 60% of its last 65 games last season.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Kalyan Nair answered

To determine if the sports team won at least half of the games it played last season, we need to know the total number of games won by the team and compare it to half of the total number of games played. Let's analyze the given statements:

Statement (1): The team won 60% of its first 65 games last season.
Statement (2): The team won 60% of its last 65 games last season.

Analyzing Statement (1):
- The team won 60% of its first 65 games.
- We can calculate the number of games won by multiplying 65 by 60%: 65 * 0.60 = 39 games won.
- However, we don't know anything about the remaining 35 games.

Analyzing Statement (2):
- The team won 60% of its last 65 games.
- We can calculate the number of games won by multiplying 65 by 60%: 65 * 0.60 = 39 games won.
- Similarly, we don't know anything about the first 35 games.

Combining both statements:
- We know that the team won 39 games in both the first 65 games and the last 65 games.
- However, we still don't know anything about the remaining 30 games from the first 65 or the remaining 30 games from the last 65.

Therefore, even when both statements are considered together, we still don't have enough information to determine if the team won at least half of the 100 games played last season.

Hence, the correct answer is option (e) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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If k is an integer greater than 1, is k equal to 2^r for some positive integer r?
(1) k is divisible by 2⁶.
(2) k is not divisible by any odd integer greater than 1.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Pioneer Academy answered

Statement (1): k is divisible by 2⁶.
If k is divisible by 2⁶, it means that k is a multiple of 2⁶. However, being divisible by 2⁶ does not guarantee that k is equal to 2r for some positive integer r. For example, if k = 2⁶, it is divisible by 2⁶, but it cannot be expressed as 2r since r would have to be 13, which is not a positive integer. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): k is not divisible by any odd integer greater than 1.
This statement tells us that k is not divisible by any odd prime number greater than 1. In other words, k does not have any odd prime factors. However, this statement also does not provide enough information to determine if k can be expressed as 2r for some positive integer r. For example, if k = 15, it is not divisible by any odd integer greater than 1, but it cannot be expressed as 2r since it does not have a factor of 2. Therefore, statement (2) alone is not sufficient to answer the question.

By evaluating both statements together, we can see that if k is divisible by 26 (statement 1) and does not have any odd prime factors (statement 2), then k must be divisible by 2 and 13. This implies that k is equal to 2 multiplied by a positive integer (r) since it has a factor of 2. Therefore, statement (2) alone is sufficient to answer the question.

Hence, the answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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When x + y is integer, is y an integer?
(1) x is an integer.
(2) x + 2y is an integer.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Kiran Nambiar answered

Understanding the Question
The question asks whether y is an integer given that x + y is an integer.
Analyzing Statement (1)
- If x is an integer, then x + y being an integer implies that y must also be an integer.
- This is because the sum of an integer (x) and another number (y) is an integer only if that other number (y) is also an integer.
Conclusion from Statement (1)
- Therefore, Statement (1) alone is sufficient to conclude that y is an integer.
Analyzing Statement (2)
- Statement (2) states that x + 2y is an integer.
- However, this does not provide direct information about y.
Possible Scenarios from Statement (2)
- If x is an integer, then 2y must also be an integer, implying y could be an integer or a half-integer (e.g., 0.5) depending on the value of x.
- If x is not an integer, the result is even less clear, as y could still be a non-integer.
Conclusion from Statement (2)
- Statement (2) alone is not sufficient to determine if y is an integer.
Combining Both Statements
- Statement (1) already established that if x is an integer, y must also be an integer.
- Statement (2) does not change or add any new information regarding y's integer status.
Final Conclusion
- Since Statement (1) alone is sufficient, while Statement (2) is not, the correct answer is option 'D': Each statement alone is sufficient to answer the question asked.

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A teacher distributed a number of candies, cookies, and toffees among the students in the class. How many students were there in the class?
(1) The numbers of candies, cookies, and toffees that each student received were in the ratio 3:4:5, respectively.
(2) The teacher distributed a total of 27 candies, 36 cookies, and 45 toffees
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Tanishq Yadav answered

Statement (1): The numbers of candies, cookies, and toffees that each student received were in the ratio 3:4:5, respectively.
From this statement alone, we know the ratio of candies, cookies, and toffees that each student received, but we don't know the total number of candies, cookies, and toffees distributed. Therefore, we cannot determine the number of students in the class. This statement alone is not sufficient to answer the question.

Statement (2): The teacher distributed a total of 27 candies, 36 cookies, and 45 toffees.
From this statement alone, we know the total number of candies, cookies, and toffees distributed, but we don't know how many of each item each student received. Therefore, we cannot determine the number of students in the class. This statement alone is not sufficient to answer the question.

Statements (1) and (2) together:
Combining both statements, we know the ratio of candies, cookies, and toffees distributed, as well as the total number of each item. We can use this information to find the number of students in the class.

Let's assume the number of students is x.

According to statement (1), the ratio of candies, cookies, and toffees that each student received is 3:4:5. Therefore, the total number of candies distributed is 3x, the total number of cookies distributed is 4x, and the total number of toffees distributed is 5x.

According to statement (2), the total number of candies, cookies, and toffees distributed is 27, 36, and 45, respectively.

Setting up the ratios:
3x/27 = 4x/36 = 5x/45

Simplifying the ratios:
x/9 = x/9 = x/9

Since the ratios are equal, we can equate them to find x:
x = 9

Therefore, there were 9 students in the class.

Both statements together are sufficient to answer the question.

Therefore, the correct answer is option (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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Byrne and some of his friends go out to dinner and spend $111, excluding tax and tip. If the group included both men and women, how many men were in the group?
(1) There are a total of five people at the table, including Byrne.
(2) The women order meals that cost an average of $19 and the men order meals that cost and average of $27.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Notes Wala answered

Statement (1): There are a total of five people at the table, including Byrne.

This statement tells us that the total number of people in the group is five, including Byrne. However, it does not provide any information about the gender distribution or how much each person spent. Therefore, we cannot determine the number of men in the group based on this statement alone.

Statement (2): The women order meals that cost an average of $19, and the men order meals that cost an average of $27.

This statement gives us information about the average cost of meals for women and men separately. However, it does not provide the total number of people in the group or the total amount spent. Without knowing the total amount spent or the number of people, we cannot determine the number of men in the group based on this statement alone.

Since neither statement alone is sufficient to answer the question, we need to consider both statements together:

By combining both statements, we know that there are five people in total (including Byrne) and that the average cost of meals for women is $19, while the average cost for men is $27. However, we still don't have enough information to determine the exact number of men in the group.

For example, it's possible that there are three women and two men in the group, or two women and three men. In both cases, the total number of people and the average costs of meals for women and men would be consistent with the given information. Therefore, even when considering both statements together, we cannot determine the number of men in the group.

Since both statements together are not sufficient to answer the question, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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What is the absolute difference between the cubes of two different non-negative integers?
(1) One of the integers is 2 greater than the other integer.
(2) The square of the sum of the integers is 49 greater than the product of the integers.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Maya Choudhury answered

Statement 1: One of the integers is 2 greater than the other integer.

This statement alone is not sufficient to determine the absolute difference between the cubes of the two integers. We can demonstrate this by considering two scenarios:

Scenario 1: Let one integer be x and the other be x + 2. In this case, the absolute difference between the cubes would be (x + 2)^3 - x^3, which cannot be simplified further.

Scenario 2: Let one integer be x and the other be x - 2. In this case, the absolute difference between the cubes would be x^3 - (x - 2)^3, which also cannot be simplified further.

Therefore, statement 1 alone is not sufficient.

Statement 2: The square of the sum of the integers is 49 greater than the product of the integers.

Let the two integers be x and y.

The square of the sum of the integers is (x + y)^2.

The product of the integers is xy.

According to the given information, (x + y)^2 = xy + 49.

Expanding (x + y)^2, we have x^2 + 2xy + y^2 = xy + 49.

Rearranging the terms, we get x^2 + y^2 - xy = 49.

This equation can be simplified further using the formula for the sum and product of roots of a quadratic equation:

(x + y)^2 - 3xy = 49.

Since we know that (x + y)^2 = xy + 49, we can substitute this into the equation:

xy + 49 - 3xy = 49.

Simplifying, we have -2xy = 0, which implies xy = 0.

From this, we can conclude that either x or y must be equal to 0.

Combining the statements:

From statement 1, we know that one of the integers is 2 greater than the other integer. Let's assume that x = y + 2.

From statement 2, we know that xy = 0.

Substituting the value of x from statement 1 into the equation xy = 0, we have (y + 2)y = 0.

Expanding, we get y^2 + 2y = 0.

Factoring out y, we have y(y + 2) = 0.

From this equation, we can see that y = 0 or y = -2.

If y = 0, then x = 2, and the absolute difference between the cubes of the integers is 2^3 - 0^3 = 8.

If y = -2, then x = 0, and the absolute difference between the cubes of the integers is 0^3 - (-2)^3 = -8.

Therefore, by combining the information from both statements, we can determine the absolute difference between the cubes of the two integers.

Hence, the correct answer is option C.

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If positive integer y is equal to the sum of all the unique factors of the positive integer x, is |x-y| > 1?
(1) x is not prime.
(2) x does not equal 1.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): x is not prime.
This statement tells us that x is a composite number, meaning it has factors other than 1 and itself. If x is composite, we can find the factors of x and calculate their sum to obtain y. However, this statement alone doesn't provide information about the specific values of x and y or whether |x - y| > 1.

For example, if x = 6, which is not prime, the factors of 6 are 1, 2, 3, and 6. The sum of these factors is y = 1 + 2 + 3 + 6 = 12. In this case, |x - y| = |6 - 12| = 6 > 1.

On the other hand, if x = 4, which is also not prime, the factors of 4 are 1, 2, and 4. The sum of these factors is y = 1 + 2 + 4 = 7. In this case, |x - y| = |4 - 7| = 3 > 1.

Statement (1) alone is not sufficient to determine whether |x - y| > 1.

Statement (2): x does not equal 1.
This statement simply tells us that x is not equal to 1. While this implies that x is a positive integer greater than 1, it doesn't provide any direct information about the factors of x, the sum of the factors, or the difference |x - y|.

For example, if x = 6, the factors of 6 are 1, 2, 3, and 6. The sum of these factors is y = 1 + 2 + 3 + 6 = 12. In this case, |x - y| = |6 - 12| = 6 > 1.

On the other hand, if x = 2, the factors of 2 are 1 and 2. The sum of these factors is y = 1 + 2 = 3. In this case, |x - y| = |2 - 3| = 1 ≤ 1.

Statement (2) alone is not sufficient to determine whether |x - y| > 1.

When considering both statements together, we have the following information:

From statement (1): x is not prime.
From statement (2): x does not equal 1.

These statements together imply that x is a composite number greater than 1. In other words, x has factors other than 1 and itself. Since y is the sum of all the unique factors of x, y will always be greater than 1.

Therefore, |x - y| will always be greater than 1.

When considering both statements together, we can determine that |x - y| > 1.

Hence, both statements together are sufficient to answer the question asked.

The correct answer is:

C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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If m and n are positive integers greater than 1, is m + n odd?
(1) m is a divisor of 16
(2) The units digit of 12^m + 28ⁿ is 6.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): m is a divisor of 16.
This statement tells us that m is a factor of 16, which means m can take on the values 1, 2, 4, 8, or 16. However, it does not provide any information about the value of n. Therefore, statement (1) alone is not sufficient to determine if m + n is odd.

Statement (2): The units digit of 12m + 28n is 6.
This statement gives us information about the units digit of the sum 12m + 28n. Since the units digit is 6, we know that the sum ends in 6. However, it does not provide any information about whether the sum is odd or even. For example, if m = 1 and n = 2, the sum 12m + 28n is 76, which is even. But if m = 1 and n = 1, the sum is 40, which is even. Therefore, statement (2) alone is not sufficient.

By considering both statements together, we have the information that m is a divisor of 16 and the units digit of 12m + 28n is 6. From statement (1), we know that m can take on the values 1, 2, 4, 8, or 16. When we consider the units digit of 12m + 28n, we can see that for all these values of m, the units digit of the sum will always be 6. However, this still does not determine whether the sum m + n is odd or even.

For example, if m = 2 and n = 4, then m + n = 6, which is even. But if m = 1 and n = 5, then m + n = 6, which is odd.

Therefore, when both statements are considered together, we still cannot determine whether m + n is odd or even. Thus, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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Of the 60 animals on a certain farm, 2/3 are either pigs or cows. How many of the animals are cows?
(1) The farm has more than twice as many cows as pigs
(2) The farm has more than 12 pigs
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Bhavana Kulkarni answered

Statement (1): The farm has more than twice as many cows as pigs
This statement tells us that the number of cows is greater than twice the number of pigs. However, it does not provide us with any specific information about the total number of animals or the ratio of cows to pigs. Therefore, this statement alone is not sufficient to answer the question asked.

Statement (2): The farm has more than 12 pigs
This statement tells us that the number of pigs on the farm is greater than 12. However, it does not provide us with any information about the number of cows or the ratio of cows to pigs. Therefore, this statement alone is not sufficient to answer the question asked.

Statements (1) and (2) together:
Combining the two statements, we know that the number of cows is greater than twice the number of pigs, and the number of pigs is greater than 12. This means that there must be at least 13 pigs on the farm.

Let's consider two scenarios:
Scenario 1: If there are 13 pigs on the farm, then the number of cows must be greater than 2 * 13 = 26. In this case, the total number of animals on the farm would be at least 13 pigs + 26 cows = 39 animals. However, this is less than the given total of 60 animals, so it is not possible.

Scenario 2: If there are more than 13 pigs on the farm, then the number of cows must be greater than 2 * (number of pigs). In this case, the total number of animals on the farm would be at least (number of pigs) + (number of cows) > 13 + 2 * (number of pigs) = 39 animals. Since we are given that there are 60 animals on the farm, this scenario is possible.

Therefore, by considering both statements together, we can conclude that the number of cows on the farm is greater than the number of pigs, but we cannot determine the exact number of cows or pigs. Hence, both statements together are sufficient to answer the question asked, but neither statement alone is sufficient. The correct answer is (C).

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Fewer than 48 children were surveyed about their pet preferences. Some of the children prefer foxes, some prefer wolfs, while the remaining prefer vultures. How many of the children surveyed prefer foxes as pets?
(1) The ratio of the number of children who prefer foxes to the number of children who prefer wolfs is 4 to 3.
(2) The ratio of the number of children who prefer foxes to the total number of children surveyed is 5 to 9.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): The ratio of the number of children who prefer foxes to the number of children who prefer wolves is 4 to 3.
This statement provides information about the ratio of children who prefer foxes to those who prefer wolves. However, it doesn't provide any specific information about the total number of children surveyed or the number of children who prefer vultures. Without knowing the total number of children surveyed or the ratio of children who prefer vultures, we cannot determine the number of children who prefer foxes. Therefore, statement (1) alone is not sufficient.

Statement (2): The ratio of the number of children who prefer foxes to the total number of children surveyed is 5 to 9.
This statement provides information about the ratio of children who prefer foxes to the total number of children surveyed. However, it doesn't provide any specific information about the number of children who prefer wolves or vultures. Without knowing the number of children who prefer wolves or vultures, we cannot determine the number of children who prefer foxes. Therefore, statement (2) alone is not sufficient.

By considering both statements together, we still cannot determine the number of children who prefer foxes. While statement (1) gives us the ratio of children who prefer foxes to those who prefer wolves, and statement (2) gives us the ratio of children who prefer foxes to the total number of children surveyed, we don't have enough information to calculate the exact numbers or ratios of children who prefer each pet preference.

Therefore, when both statements are considered together, we still don't have enough information to determine the number of children surveyed who prefer foxes. Additional data is needed.

Hence, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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If the integers a and n are greater than 1 and the product of the first 8 positive integers is a multiple of aⁿ, what is the value of a ?
(1) aⁿ = 64
(2) n=6
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): an = 64
This statement tells us that the product of the first 8 positive integers is equal to 64. However, it does not provide any information about the value of n. Therefore, statement (1) alone is not sufficient to determine the value of a.

Statement (2): n = 6
This statement provides a specific value for n, which is 6. However, it does not give any information about the product of the first 8 positive integers or the value of a. Therefore, statement (2) alone is not sufficient to determine the value of a.

By considering both statements together, we have the information that n = 6 and an = 64. Substituting the value of n into the equation, we get a^6 = 64. Taking the sixth root of both sides, we find a = 2.

Therefore, statement (2) alone is sufficient to determine the value of a. Thus, the answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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If 8x > 4 + 6x, what is the value of the integer x?
(1) 6 – 5x > -13
(2) 3 – 2x < -x + 4 < 7.2 – 2x
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered

Statement 1: 6 – 5x > -13
This statement provides an additional inequality. By solving this inequality, we can determine the range of values for x. Solving it:
6 - 5x > -13
Add 5x to both sides:
6 > -13 + 5x
19 > 5x
Divide both sides by 5 (since the inequality sign doesn't change when dividing by a positive number):
19/5 > x
x < 19/5

Statement 1 alone is sufficient to determine a range for x, but it doesn't provide an exact value for x.

Statement 2: 3 – 2x < -x + 4 < 7.2 – 2x
This statement provides a compound inequality. By solving this compound inequality, we can determine the range of values for x. Solving it:
3 - 2x < -x + 4 < 7.2 - 2x
We can simplify it by subtracting x from all parts of the inequality:
3 - 3x < 4 < 7.2 - 3x
Simplify further:
-3x + 3 < 4 < -3x + 7.2
Now we have two separate inequalities:
-3x + 3 < 4
4 < -3x + 7.2

Solving the first inequality:
-3x < 1
Divide by -3 (remember to reverse the inequality sign since we're dividing by a negative number):
x > -1/3

Solving the second inequality:
4 < -3x + 7.2
-3x < 3.2
Divide by -3 (reverse the inequality sign):
x > -3.2/3

Combining the two inequalities, we have:
x > -1/3 and x > -3.2/3

To find the common range of values for x, we take the greater of the two lower bounds, which is x > -1/3.

Statement 2 alone is sufficient to determine a range for x, but it doesn't provide an exact value for x.

Considering both statements together:
From statement 1, we know that x < 19/5, which gives us an upper bound for x.
From statement 2, we know that x > -1/3, which gives us a lower bound for x.

Combining the information, we have:
-1/3 < x < 19/5

Therefore, with both statements together, we have a range for x but not an exact value.

The answer is option D: EACH statement ALONE is sufficient to answer the question asked.

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If n and p are positive integers, is the ratio of n to p 2 : 1?
(1) n – p < 0
(2) np = 20
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): n - p < 0
This statement indicates that n is less than p. However, it doesn't provide direct information about the specific ratio between n and p. We cannot determine if the ratio is 2:1 based on this statement alone.

Statement (2): np = 20
This statement tells us that the product of n and p is 20. Without further information, we cannot determine the individual values of n and p or their ratio. For example, n could be 4 and p could be 5, resulting in a ratio of 4:5, which is not 2:1. Alternatively, n could be 2 and p could be 10, resulting in a ratio of 2:10, which simplifies to 1:5 and is also not 2:1. Statement (2) alone is not sufficient.

Combining both statements:
From statement (1), we know that n - p < 0, which implies that n is less than p. From statement (2), we know that np = 20. Combining these statements, we can deduce that the only possible combination of positive integers n and p that satisfies both conditions is n = 2 and p = 10. In this case, the ratio of n to p is indeed 2:1. Therefore, by considering both statements together, we can determine that the ratio of n to p is 2:1.

Hence, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

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Both a, b, and c are 3-digits integers, where a=b+c. Is the hundreds' digit of number a equal to sum of that of b and c?
(1) Tens' digit of a=tens' digit of b+tens' digit of c
(2) Units' digit of a=units' digit of b+units' digit of c
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Pioneer Academy answered

To determine whether the hundreds' digit of number "a" is equal to the sum of the hundreds' digits of numbers "b" and "c," let's evaluate the two given statements:

Statement (1): Tens' digit of "a" = tens' digit of "b" + tens' digit of "c."

This statement alone does not provide any information about the hundreds' digit. The tens' digit and hundreds' digit are independent of each other, so we cannot conclude whether the hundreds' digit of "a" is equal to the sum of the hundreds' digits of "b" and "c" based on this statement alone.

Statement (2): Units' digit of "a" = units' digit of "b" + units' digit of "c."

Similar to statement (1), this statement also does not provide any information about the hundreds' digit of the numbers. The units' digit and hundreds' digit are unrelated, so we cannot determine whether the hundreds' digit of "a" is equal to the sum of the hundreds' digits of "b" and "c" based on this statement alone.

Since neither statement alone provides the required information, we need to evaluate both statements together.

By considering both statements, we can determine the relationship between the hundreds' digit of "a" and the hundreds' digits of "b" and "c." However, we do not have any information about the tens' and units' digits, which are necessary to solve the problem.

Therefore, the correct solution is (A): Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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If x, y, and d are integers and d is odd, are both x and y divisible by d ?
(1) x + y is divisible by d.
(2) x − y is divisible by d.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Pioneer Academy answered

To analyze the given statements and determine whether both x and y are divisible by d, let's examine each statement individually:

Statement (1): x + y is divisible by d.
This means that the sum of x and y is divisible by d.
However, we don't have any information about the individual values of x and y or their divisibility by d.
For example, if x = 3, y = 2, and d = 5, the sum x + y is divisible by d, but neither x nor y is individually divisible by d.
Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): x - y is divisible by d.
Similarly, this statement tells us that the difference between x and y is divisible by d.
However, just like in statement (1), we lack information about the divisibility of x and y individually.
For instance, if x = 7, y = 4, and d = 3, the difference x - y is divisible by d, but neither x nor y is divisible by d.
Hence, statement (2) alone is not sufficient to answer the question.

Combining both statements, we have the information that both the sum (x + y) and the difference (x - y) are divisible by d.
This implies that both x and y must be divisible by d.
For example, if x = 6, y = 3, and d = 3, both (x + y) and (x - y) are divisible by d, and x and y are individually divisible by d.
Thus, together, statements (1) and (2) are sufficient to answer the question.

Therefore, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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If x is a positive integer, is x a prime number?
(A) x - p = q - x = k, where p, q, and k are prime numbers.
(B) The total odd factor of 15k³ is 4, where k is a prime number.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

Let's analyze each statement separately:

Statement (1): x - p = q - x = k, where p, q, and k are prime numbers.

This statement provides an equation involving x and prime numbers p, q, and k. However, it doesn't give us any specific information about the value of x or whether x itself is a prime number. Therefore, statement (1) alone is not sufficient to determine whether x is a prime number.

Statement (2): The total odd factor of 15k³ is 4, where k is a prime number.

This statement provides information about the total odd factors of a specific expression involving the prime number k. However, it doesn't provide any direct information about the value of x or whether x is a prime number. Therefore, statement (2) alone is not sufficient to determine whether x is a prime number.

Now let's consider both statements together. Even when considering both statements together, we still don't have any direct information about the value of x or whether x is a prime number. The statements provide information about prime numbers in different contexts, but they don't provide any direct information about x itself.

Therefore, when we consider both statements together, we still cannot determine whether x is a prime number. Hence, the answer is E: Statements (1) and (2) together are NOT sufficient to answer the question asked, and additional data are needed.

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If b is prime and the symbol # represents one of the following operations: addition, subtraction, multiplication, or division, is the value of b # 2 even or odd?
(1) (b # 1) # 2 = 5
(2) 4 # b = 3 # (1 # b) and b is even
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Talent Skill Learning answered

Statement (1): (b # 1) # 2 = 5
This statement tells us that the result of performing the operation represented by # on b and 1, and then performing the operation on the result and 2, is equal to 5. However, since we don't know the specific operation represented by #, we can't determine the value of b # 2. Therefore, statement (1) alone is not sufficient to determine whether the value of b # 2 is even or odd.

Statement (2): 4 # b = 3 # (1 # b) and b is even
This statement tells us that the result of performing the operation represented by # on 4 and b is equal to the result of performing the operation represented by # on 3 and the result of performing the operation on 1 and b. Additionally, it specifies that b is even. However, similar to statement (1), since we don't know the specific operation represented by #, we can't determine the value of b # 2. Therefore, statement (2) alone is not sufficient to determine whether the value of b # 2 is even or odd.

Since neither statement alone is sufficient, let's consider them together. Unfortunately, even when considering both statements together, we still don't have enough information to determine the value of b # 2. The relationship between b, the operation represented by #, and the specific values involved in the statements is not clear. Therefore, statements (1) and (2) together are not sufficient to determine whether the value of b # 2 is even or odd.

As a result, each statement alone is not sufficient to answer the question asked. The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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For integers x, b and c, x² + bx + c = 0. Is x > 0?
(1) b > 0
(2) c > 0
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

To determine if x > 0, we need to consider the discriminant of the quadratic equation x² + bx + c = 0.

The discriminant is given by Δ = b² - 4ac. If Δ > 0, the quadratic equation has two distinct real roots. If Δ = 0, it has one real root (a repeated root). If Δ < 0, it has no real roots.

Let's evaluate each statement:

Statement 1: b > 0
This statement alone does not provide information about the discriminant or the sign of x. It tells us that the coefficient b is positive, but we don't have enough information to determine the sign of x. Statement 1 is not sufficient.

Statement 2: c > 0
Similar to statement 1, this statement alone does not provide information about the discriminant or the sign of x. It tells us that the constant term c is positive, but we still cannot determine the sign of x. Statement 2 is not sufficient.

Considering both statements together:
We know that b > 0 and c > 0. However, even with this combined information, we still don't have enough information to determine the value of the discriminant Δ or the sign of x. It is possible to have cases where both b and c are positive, yet Δ < 0, leading to no real roots. Therefore, both statements together are not sufficient.

Since neither statement alone is sufficient and both statements together are also not sufficient, the answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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The revenue R, in dollars, that is generated by selling “x” units of a certain product is given by 9x² + bx + c, where x > 0 and b and c are constants. Find the value of b.
(1) The revenue generated by selling 200 units of the product is 54,0000.
(2) The roots of the equation ax² - bx + c are 20 and 9.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Abhishek Choudhury answered

A certain product is given by the formula R = 10p, where p is the price of the product in dollars.

If the price of the product is $5, then the revenue generated would be:

R = 10(5) = $50

Therefore, if the price of the product is $5, the revenue generated would be $50.

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On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?
(1) xyz < 0
(2) xy < 0
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Wizius Careers answered

Statement 1: xyz < 0
This statement tells us that the product of x, y, and z is negative. However, this statement does not provide any specific information about the relative positions of x, y, and z on the number line. It is possible that z lies between x and y, or it is also possible that z does not lie between x and y. Therefore, statement 1 alone is not sufficient to answer the question.

Statement 2: xy < 0
This statement tells us that the product of x and y is negative. Similar to statement 1, this statement does not provide any information about the position of z relative to x and y on the number line. It is also not sufficient to answer the question.

Since neither statement alone provides enough information to determine if z lies between x and y on the number line, the answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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If a and b are positive integers, what is the digit at the unit's place of 18^{(2a + 5b)}?
(1) a is even, b is a multiple of four.
(2) b = 12
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Wizius Careers answered

To determine the digit at the unit's place of 18^{2a + 5b}, we need to consider the cyclicity of the units digit of the number 18.

The units digit of any power of 18 repeats in cycles of four: 8, 4, 2, 6, 8, 4, 2, 6, and so on.

Let's analyze each statement separately:
Statement (1): a is even, b is a multiple of four.
If a is even, 2a is also even. Since the exponent of 18 is 2a + 5b, it will always be even since it contains an even number (2a) and a multiple of four (5b).

In this case, the units digit of 18^{2a + 5b} will always be 4 because the exponent is always even. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2): b = 12

If b = 12, we know the value of b but not the value of a. Without knowing the value of a, we cannot determine the value of 2a + 5b and, therefore, cannot determine the units digit of 18^{2a + 5b}. Thus, statement (2) alone is not sufficient to answer the question.

Since statement (1) alone is sufficient to answer the question, but statement (2) alone is not, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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If a and b are prime numbers, is ab even?
(1) The sum of a and b is prime
(2) The difference of a and b is prime
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): The sum of a and b is prime.
If a and b are prime numbers, their sum will only be prime if one of the numbers is 2 (the only even prime number) and the other number is odd. In this case, ab will be even since one of the factors (2) is even.

However, if both a and b are odd prime numbers, their sum will be even, but the product ab will also be odd.

Therefore, statement (1) alone is sufficient to determine that ab is even if the sum of a and b is prime, but it does not definitively determine that ab is even if the sum of a and b is not prime.

Statement (2): The difference of a and b is prime.
The difference between two prime numbers can be either even or odd. For example, if a = 5 and b = 2, their difference is 5 - 2 = 3, which is prime. In this case, ab = 5 * 2 = 10, which is even.

However, if a = 5 and b = 3, their difference is 5 - 3 = 2, which is prime. In this case, ab = 5 * 3 = 15, which is odd.

Therefore, statement (2) alone is not sufficient to determine whether ab is even or not.

Since statement (1) alone is sufficient to determine that ab is even if the sum of a and b is prime, but statement (2) does not provide enough information, the correct answer is:

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Integers x and y are both positive, and x > y. How many different committees of y people can be chosen from a group of x people?
(1) The number of different committees of x-y people that can be chosen from a group of x people is 3,060.
(2) The number of different ways to arrange x-y people in a line is 24.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Notes Wala answered

The question asks for the number of different committees of y people that can be chosen from a group of x people, given that x > y.

Statement (1) alone: The number of different committees of x-y people that can be chosen from a group of x people is 3,060.

This statement provides information about the number of committees of x-y people, but it does not directly provide information about the number of committees of y people. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) alone: The number of different ways to arrange x-y people in a line is 24.

This statement provides information about arranging x-y people in a line, but it does not provide information about committees or the number of people in the committees. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together:

Let's assume the number of people in the group is x and the number of people in the committee is y.

Statement (1) tells us that the number of committees of x-y people that can be chosen from a group of x people is 3,060. This can be represented as xC(x-y) = 3,060.

Statement (2) does not provide direct information about committees or the number of people in the committees, so it does not directly contribute to solving the problem.

By combining the two statements, we have:

xC(x-y) = 3,060

Since x > y, we know that x-y is positive.

We need to find the value of y, which represents the number of people in the committee.

Since we have only one equation and two variables (x and y), we cannot uniquely determine the values of x and y. Therefore, statements (1) and (2) together are not sufficient to answer the question.

Based on the analysis above, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Josh has a big drawer full of exactly 40 packets, each containing a marker which is either black or blue or red in colour. All 40 packets are sealed and are completely identical from outside. It is not possible to know which colour marker is inside unless the packet is fully opened. Out of 40, 23 packets contain a black marker each. How many packets contain a blue marker?
(1) Drawer has less packets containing blue markers than those containing red markers.
(2) If Josh withdraws packets without looking at their contents, he needs to draw minimum 20 packets to ensure that he has exactly 8 markers of any single colour out of red, blue, black.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

BT Educators answered

Statement 1: The drawer has fewer packets containing blue markers than those containing red markers.
This statement gives us a comparison between the number of packets containing blue markers and those containing red markers. However, it doesn't provide any specific information about the number of packets containing blue markers or their relationship with the packets containing black markers. Therefore, statement 1 alone is not sufficient to determine the number of packets containing a blue marker.

Statement 2: If Josh withdraws packets without looking at their contents, he needs to draw a minimum of 20 packets to ensure that he has exactly 8 markers of any single color out of red, blue, and black.
This statement provides information about the minimum number of packets Josh needs to draw to ensure he has exactly 8 markers of any single color. However, it doesn't directly give us information about the number of packets containing a blue marker. Therefore, statement 2 alone is not sufficient to determine the number of packets containing a blue marker.

When we consider both statements together, we still don't have enough information to determine the exact number of packets containing a blue marker. Statement 1 tells us that there are fewer packets containing blue markers than those containing red markers, but it doesn't provide any information about the number of packets containing black markers. Statement 2 gives us information about the minimum number of packets Josh needs to draw to ensure 8 markers of any single color, but it doesn't specify the distribution of colors within those packets.

Without additional information about the relationship between the packets containing black markers and the packets containing blue markers, we cannot determine the exact number of packets containing a blue marker.

Therefore, the answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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If a = |b - 6| + |b + 2|, what is the value of a?
(1) a is an integer greater than 7
(2) -2 < b < 6
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Prateek Gupta answered

From the given equation, we can see that a is the product of two absolute values, |b - 6| and |b + 2|.

To find the value of a, we need to know the values of b and the signs of b - 6 and b + 2.

Statement 1: a is an integer greater than 7.
This statement alone does not provide enough information to determine the values of b or the signs of b - 6 and b + 2. Insufficient.

Statement 2: -2 < b="" />< />
This statement provides the range of possible values for b. However, it does not provide the specific values of b or the signs of b - 6 and b + 2. Insufficient.

Combining both statements, we still do not have enough information to determine the values of b or the signs of b - 6 and b + 2. Insufficient.

Therefore, the answer is E: Both statements together are still insufficient to answer the question.

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What is the units digit of the three-digit integer N?
(1) When N is rounded to the nearest hundred, the result is 50 less than the result when N is rounded to the nearest ten
(2) N is divisible by 4.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Wizius Careers answered

Statement (1): When N is rounded to the nearest hundred, the result is 50 less than the result when N is rounded to the nearest ten.

This statement implies that the tens digit of N is greater than or equal to 5. For example, if N is rounded to the nearest hundred and the result is 100, then when rounded to the nearest ten, the result must be 150.

However, this statement alone does not provide enough information to determine the units digit of N.

Statement (2): N is divisible by 4.

A number is divisible by 4 if its last two digits form a number that is divisible by 4. For example, if N ends in 24, 28, 32, etc., it is divisible by 4.

Knowing that N is divisible by 4 helps us narrow down the possible values for the units digit. However, it still doesn't give us a unique answer.

Combining both statements:

From statement (1), we know that the tens digit of N is greater than or equal to 5. From statement (2), we know that N is divisible by 4, meaning the last two digits must form a number divisible by 4.

Combining these conditions, we can list the possibilities for the tens and units digits:

54 (divisible by 4)
64 (divisible by 4)
74 (not divisible by 4)
84 (divisible by 4)
94 (not divisible by 4)

Only the numbers 54, 64, and 84 satisfy both conditions. Therefore, the units digit of N could be 4, 6, or 8.

Since we can't determine the units digit with certainty, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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John, Paul and Linda drove on a 1,500 mile trip. If they shared the driving, which of the three drove the longer distance
(1) John drove one hour longer than Paul but at an average race of 5 miles per hour slower than Paul.
(2) Linda drove 9 hours and averaged 50 miles per hour.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Pioneer Academy answered

Statement (1): John drove one hour longer than Paul but at an average race of 5 miles per hour slower than Paul.

Statement (2): Linda drove 9 hours and averaged 50 miles per hour.

From statement (1), we know that John drove one hour longer than Paul. However, we don't have any information about the distance they each drove. Additionally, we are told that John's average speed was 5 miles per hour slower than Paul's, but we don't know either John's or Paul's average speeds. Therefore, statement (1) alone is not sufficient to determine which of the three drove the longer distance.

From statement (2), we know that Linda drove for 9 hours and averaged 50 miles per hour. This gives us the distance Linda drove, which is 9 hours * 50 miles per hour = 450 miles. However, this information alone does not help us determine whether John or Paul drove a longer distance. Therefore, statement (2) alone is not sufficient.

Combining the information from both statements, we still don't have enough information to determine which of the three drove the longer distance. We have information about Linda's distance but lack information about John's and Paul's distances. Therefore, both statements together are not sufficient.

Hence, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?
(1) #(10k) = 10k
(2) #(100k) is 10300.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): #(10k) = 10k
This statement tells us that when the number 10k is rounded to the nearest integer, it is equal to 10k. However, it doesn't provide any specific information about the value of k or the rounding behavior. Without knowledge of the value of k, we cannot determine the units digit of #k based on this statement alone.

Statement (2): #(100k) is 10300.
This statement tells us that when the number 100k is rounded to the nearest integer, it is equal to 10300. From this statement, we can deduce that k lies between two consecutive integers such that rounding 100k to the nearest integer results in 10300. However, this information alone doesn't directly provide the value of k or the units digit of #k.

By evaluating both statements together, we know that #(10k) = 10k from statement (1), and #(100k) = 10300 from statement (2). Combining these statements doesn't provide additional information that helps us determine the units digit of #k. The relationship between #(10k) and #(100k) is not clear, and we still lack specific information about the value of k.

Therefore, statement (2) alone is sufficient to determine the units digit of #k, but statement (1) alone is not sufficient. The correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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The symbols $ and # each represent one of the following operations: addition, subtraction, multiplication or division. What is the value of 1 $ 1 # 1?
(1) 2 $ 2 # 2 = 3
(2) 3 $ 2 # 1 = 5
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): 2 $ 2 # 2 = 3 From this statement, we know the result of the expression 2 $ 2 # 2 is 3. However, this information alone doesn't provide enough insight into the operations $ and # or how they interact. Therefore, statement (1) alone is not sufficient to determine the value of 1 $ 1 # 1.

Statement (2): 3 $ 2 # 1 = 5 From this statement, we know the result of the expression 3 $ 2 # 1 is 5. Similar to statement (1), this information alone doesn't provide enough information to deduce the operations $ and # or their impact on the value of the expression 1 $ 1 # 1. Thus, statement (2) alone is not sufficient.

Combining both statements: By combining both statements, we have the information that 2 $ 2 # 2 equals 3 and 3 $ 2 # 1 equals 5. However, this still doesn't give us a clear understanding of the specific operations represented by $ and #. Therefore, even when considering both statements together, we cannot determine the value of 1 $ 1 # 1.

As a result, the correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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If a and b are positive integers, what is the digit at the unit's place of 18^(2a + 5b)?
(1) a is even, b is a multiple of four.
(2) b = 12
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): a is even, b is a multiple of four.
If a is even, then 2a is a multiple of 4. Additionally, if b is a multiple of four, then 5b is also a multiple of 4. Since both 2a and 5b are multiples of 4, the exponent 2a + 5b is a multiple of 4. As a result, 18^{(2a + 5b)} will end in the same digit as 18⁴, as the unit's digit repeats every 4 powers of 18.

The unit's digit of 18⁴ is 4. Therefore, statement (1) alone is sufficient to determine the digit at the unit's place.

Statement (2): b = 12
This statement gives the value of b as 12. However, it does not provide any information about a or the exponent 2a + 5b. Without knowing the value of a, we cannot determine the digit at the unit's place. Therefore, statement (2) alone is not sufficient.

Since statement (1) alone is sufficient to determine the digit at the unit's place, but statement (2) alone is not sufficient, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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On a fishing expedition, a group of 13 fishermen caught a total of 160 fish. \ Did any one fisherman catch more than 15 fish?
(1) The fisherman who caught the third-most fish caught 11 fish.
(2) The fisherman who caught the second-most fish caught 12 fish
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

BT Educators answered

Statement (1): The fisherman who caught the third-most fish caught 11 fish.
This statement alone does not provide enough information to determine whether any one fisherman caught more than 15 fish. We don't know how many fish the first and second most productive fishermen caught, so we cannot make a definitive conclusion.

Statement (2): The fisherman who caught the second-most fish caught 12 fish.
This statement alone also does not provide enough information to determine whether any one fisherman caught more than 15 fish. We still don't know the number of fish caught by the most productive fisherman or any other fishermen besides the second-most productive one.

By combining both statements, we can determine the answer. Since the fisherman who caught the third-most fish caught 11 fish, and the fisherman who caught the second-most fish caught 12 fish, we know that the most productive fisherman must have caught more than 15 fish. Therefore, each statement alone is sufficient to answer the question.

Hence, the correct answer is (D): EACH statement ALONE is sufficient to answer the question asked.

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If Rebeca drives to work at x mph she will be one minute late, but if she drives at y mph she will be one minute early. How far (in miles) does Rebeca drive to work?
(1) x and y differ by seven miles per hour.
(2) y is 11% greater than x.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Pioneer Academy answered

The given problem involves determining the distance Rebeca drives to work based on her driving speed.

Let's analyze each statement separately:

Statement (1): x and y differ by seven miles per hour.

This statement provides the difference in speeds between x and y, but it doesn't provide any specific information about their values or the relationship between them. We cannot determine the distance based on this statement alone.

Statement (2): y is 11% greater than x.

This statement gives us a specific relationship between x and y, stating that y is 11% greater than x. However, without knowing the actual values of x or y, we cannot determine the distance Rebeca drives.

Considering both statements together:

When we combine the information from both statements, we can deduce some additional information. Let's say x represents Rebeca's driving speed in miles per hour. From statement (1), we can infer that y is x + 7 mph. Furthermore, statement (2) tells us that y is 11% greater than x. Mathematically, this can be expressed as y = x + 0.11x = 1.11x.

Now, we can set up equations based on the given time differences:

Equation 1: Distance / x = Time (One minute late)
Equation 2: Distance / y = Time (One minute early)

Since the distance remains the same in both equations, we can equate them:

Distance / x = Distance / y

Cross-multiplying, we get:

Distance * y = Distance * x

Since y = 1.11x (from statement 2), we can substitute it into the equation:

Distance * 1.11x = Distance * x

Dividing both sides by Distance and canceling out the common factor of Distance, we have:

1.11x = x

Simplifying further, we get:

0.11x = 0

This equation implies that x must be 0, which doesn't make sense in the context of the problem. Therefore, this scenario is not possible.

Hence, the combination of both statements (1) and (2) together is not sufficient to determine the distance Rebeca drives to work. Therefore, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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