Test: Number Properties - GMAT MCQ

Statement 1: The tens digit of a is equal to the units digit of b, and the tens digit of b is equal to the units digit of a
This means the numbers will be of the form xy and yx.

On adding, we get (10x + y) + (10y + x) = 11x + 11y
This clearly is divisible by 11
SUFFICIENT

Statement 2: Both a and b are odd
The different values of a and b can be:
a = 23 and b = 35
Here a + b = 58. Not a multiple of 11

a = 11 and b = 33
a + b = 44. Multiple of 11
INSUFFICIENT

Statement 1: x is a positive interger. Meaning X > 0 and is an Integer.
Now, (6)^P (where P is a positive Integer) will always give the last digit as 6 and hence remainder will always remain 6. → SUFFICIENT

Statement 2: This is just a filtered part of Statement 1. Meaning in Statement 1, I already said we will get a single solution for X > 0 and X as an integer. Now, this includes both even and Odd. This, Statement 2 is also sufficient.
Hence, D.

Statement 1: j + 1 is not prime
There are several values of j that satisfy statement 1. Here are two:
Case a: j = 3, in which case j+1 = 4 and 4 is not prime. In this case j IS prime
Case b: j = 9, in which case j+1 = 10 and 10 is not prime. In this case j is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: j is odd
There are several values of j that satisfy statement 1. Here are two:
Case a: j = 3 (and 3 is odd). In this case j IS prime
Case b: j = 9 (and 9 is odd). In this case j is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of j that satisfy BOTH statements. Here are two:
Case a: j = 3, in which case j is odd AND j+1 is not prime. In this case j IS prime
Case b: j = 9, in which case j is odd AND j+1 is not prime. In this case j is NOT prime
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Statement (1): |u| + |v| = 0
The sum of the absolute values of u and v is equal to 0. Since absolute values are always non-negative, for their sum to be 0, both u and v must be 0. Therefore, u = 0. Statement (1) alone is sufficient to determine the value of u.

Statement (2): u² + v² = 0
The sum of the squares of u and v is equal to 0. Since squares are always non-negative, for their sum to be 0, both u² and v² must be 0. This means both u and v must be 0. Therefore, u = 0. Statement (2) alone is also sufficient to determine the value of u.

Since both statement (1) and statement (2) individually provide sufficient information to determine the value of u, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

Step 1: Apply Variable Approach(VA)
Step II: After applying VA, if C is the answer, check whether the question is key questions.
Step III: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).
Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.

Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).
To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]
Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.
We have to find Is A/B is an integer? where A > B > 0

Second and the third step of Variable Approach: From the original condition, we have 2 variables (A and B). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 2 equations, C would most likely be the answer.
But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A or B.

Let’s take a look at each condition.
Condition(1) tells us that Neither A nor B is an integer.
⇒ If A = 1/3 and B = 1616 then A/B = 6/3 = 2 is an integer - YES
⇒ But if A = 1/3 and B = 1/2 then A/B = 2/3 = is not an integer - NO
Since the answer is not a unique YES or NO, the condition is not sufficient by CMT 1.
Condition(2) tells us that A = 3B.
⇒ A/B = 3 is an integer - YES
Since the answer is unique YES, condition(2) alone is sufficient by CMT 1.
Condition (2) alone is sufficient.

So, B is the correct answer.

Statement (1): Set X contains 150 even integers.
From this statement, we know that out of the 600 elements in Set X, 150 are even integers. However, this statement doesn't provide any information about the odd integers or their distribution. We cannot determine the exact number of positive odd integers in Set X based on this statement alone. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): 70% of the odd integers in Set X are positive.
This statement tells us that 70% of the odd integers in Set X are positive. However, we don't have any information about the total number of odd integers in Set X or the total number of integers in Set X. Without knowing these quantities, we cannot determine the number of positive odd integers. Thus, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still cannot determine the exact number of positive odd integers in Set X. Statement (1) provides information about even integers, while statement (2) gives information about the percentage of positive odd integers but not their actual count. Without more specific data, we cannot answer the question.

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

Statement (1): 14 < n < 20
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. There are a total of 8 prime numbers in this range.

Statement (2): 13 < n < 17
The prime numbers less than 17 are 2, 3, 5, 7, 11, and 13. There are a total of 6 prime numbers in this range.

Since statement (2) provides a narrower range for n (13 < n < 17) compared to statement (1) (14 < n < 20), statement (2) alone is sufficient to determine the number of positive prime numbers less than n. Therefore, the answer is option B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement 1
(1) The units digit of y² is 1
The unit digit of y can be either 1 or the unit digits can be 9.
The statement is not sufficient to answer the question.
Hence, we can eliminate A and D.

Statement 2
(2) The units digit of y does not equal 1
Not sufficient, as the unit digit can be any integer between 2 and 9, both inclusive.
We can eliminate B.

Combined
From statements 1 and 2, we can conclude that the unit digit = 9.
Sufficient.

Statement (1) <x> can take multiple values: 1(0 < x ≤ 1), 2(1 < x ≤ 2) and so on. Not sufficient.
Statement (2) <x> can take multiple values: 1(0 < x ≤ 1), 0(−1 < x ≤ 0) and so on. Not sufficient.
Combined (1) and (2) taken together will result in 0 < x < 1. We can find the unique value of <x>, which is 11. Sufficient.

To determine whether k³ > k, let's analyze each statement:

Statement (1): k is not negative
If k is not negative, it means k can be zero or a positive number. We can test this statement with examples:

• For k = 0, we have 0³ = 0, which is not greater than 0. So, k³ is not greater than k in this case.
• For k = 1, we have 1³ = 1, which is equal to 1. So, k³ is not greater than k in this case.
• For k = 2, we have 2³ = 8, which is greater than 2. So, k³ is greater than k in this case.

From the examples above, we can see that statement (1) alone is not sufficient to determine whether k³ > k.

Statement (2): k is prime
If k is prime, it means k is a positive integer greater than 1 that has no divisors other than 1 and itself. We can test this statement with examples:

• For k = 2, we have 2³ = 8, which is greater than 2. So, k³ is greater than k in this case.
• For k = 3, we have 3³ = 27, which is greater than 3. So, k³ is greater than k in this case.
• For k = 4, we have 4³ = 64, which is greater than 4. So, k³ is greater than k in this case.

From the examples above, we can see that statement (2) alone is sufficient to determine that k³ > k.

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