If Z is a positive integer such that
Steps 1 & 2: Understand Question and Draw Inferences
Given:
We need to find the value of Y.
Step 3: Analyze Statement 1 independently
Not sufficient to determine a unique value of Y.
Step 4: Analyze Statement 2 independently
(2) Y1 < 4
Multiple values of Y possible. Not sufficient.
Step 5: Analyze Both Statements Together (if needed)
So, even after combining both statements, we have 2 possible values of Y
Since we couldn’t find a unique value of Y, the correct answer. Is Option E.
Simplify the given expression
Correct Answer: Option B
What is the value of x if
Working Out:
Equating the powers of base 2 on both sides:
Equating the powers of base 3 on both sides:
The value of x that satisfies both (1) and (2) is:
x = 3
Correct Answer: Option D
If , what is the value of x?
Given:
To find: Value of x
Working Out:
Looking at the answer choices, we see that the correct answer is Option A
If x and y are nonzero numbers, what is the value of y?
Steps 1 & 2: Understand Question and Draw Inferences
Given: x ≠ 0, y ≠ 0
To find: y = ?
Step 3: Analyze Statement 1 independently
Equating the powers of 2 on both sides:
2x = 6 – 4y
x + 2y = 3
1 Linear Equation with 2 unknowns. Not sufficient to find a unique value of y.
Step 4: Analyze Statement 2 independently
Not sufficient to find a unique value of y.
Step 5: Analyze Both Statements Together (if needed)
Substituting (2) in (1):
Not sufficient to obtain a unique value of y.
Answer: Option E
If x > 0 and , what is the value of x?
Given
Approach
Working Out
As x> 0, the only possible value of x = 2.
Answer: D
If x and y are positive integers, what is the remainder when y is divided by x?
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Value of r for which we need to find the value of x and y.
Step 3: Analyze Statement 1 independently
Sufficient to answer.
Step 4: Analyze Statement 2 independently
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
If a, b and x are integers such that , what is the value of a  b
Steps 1 & 2: Understand Question and Draw Inferences
Possible values of a – b
So, we need to find the unique value of a to find the value of a – b.
Step 3: Analyze Statement 1 independently
(1) a^{3} b^{7} > 0
Sufficient to answer
Step 4: Analyze Statement 2 independently
(2) a + b > 0
Hence, we have a unique answer, where a =1 and b = 1
Thus a – b = 1 – 1 = 0.
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
A function for positive integers x and y. Is F(a, b) > a, where a and b are positive integers?
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
Using the definition of this function, we can write:
So as per Statement 1:
As a is a positive integer,a^{a} > 0. So, we can divide both sides of the inequality by a^{a} without changing the sign of the inequality.
So, b > a or b = a.
Hence, we can not say for sure if b > a. Insufficient to answer.
Step 4: Analyze Statement 2 independently
Using the definition of this function, we can write:
So, as per statement2,
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step – 4, this step is not required.
Answer: B
Is the sum of x^{y} and y^{x} positive?
(1) xy > 0
(2) x + y > 0
Steps 1 & 2: Understand Question and Draw Inferences
To Find: If x^{y} + y^{x} > 0
Step 3: Analyze Statement 1 independently
(1) xy > 0
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) x + y > 0
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Statement1 tells us that x and y have the same signs. Following cases are possible:
The only possible case is when x, y > 0 and hence x^{y} + y^{x} > 0
Sufficient to answer.
Answer: C
What is the remainder obtained when 10^{10} + 10^{5} – 24 is divided by 36?
Given:
To find: The remainder when 10^{10} + 10^{5} – 24 is divided by 36
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option E
The distance between celestial bodies is expressed in terms of light years, where 1 light year is the distance travelled by light in one year and is equal to 9.46 X 10^{17 }centimetres. If light takes 499 seconds to travel from the Sun to the Earth, which of the following is the closest to the distance, in million kilometres, between the Earth and the Sun?
Given:
To find: The option that is closest to the distance, in million kilometers, between Sun and Earth
Approach:
Working Out:
Therefore, 1.25*10^{8} kilometers =
Looking at the answer choices, we see that the correct answer is Option A
James deposited $1,000 each in two investment schemes X and Y. Scheme X doubles the invested amount every 7 years and scheme Y doubles the invested amount every 14 years. If James withdraws $500 from scheme X at the end of every 7th year, how many years will it take for the total amount invested in schemes X and Y to amount more than $40,000?
Given
To Find: Number of years it will take total amount deposited in schemes X and Y to grow to > $40,000?
Approach
Working Out
Answer: C
If x is a positive integer less than 100 such that x is divisible by 2^{y}, where y is a positive integer, what is the value of y?
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Unique value of y
Step 3: Analyze Statement 1 independently
However, we do not know if 2 is the only prime factor or x. So, we cannot find a unique value of y.
Insufficient to answer.
Step 4: Analyze Statement 2 independently
Using (2), along with the inference that y must be even, we have y = 2 as the only possible option.
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step 4, this step is not required.
Answer: B
Does a^{b} lie between 0 and 1, exclusive?
(1) a < 1
(2) b is a positive odd integer
Steps 1 & 2: Understand Question and Draw Inferences
Given: Two numbers a and b
To Find: Is 0 < a^{b} < 1
Following cases are possible:
2. 0 < a < 1 → Irrespective of the value of b, 0 < a^{b} < 1 So, the answer to the question is YES
3. If a = 0 → Irrespective of the value of b, a^{b} = 0. So, the answer to the question is NO.
4. 1 < a < 0 →
5. a ≤ 1 →
So, the answer to the question will be YES, if:
And the answer to the question will be NO, if:
So, we can answer the given question with a unique answer for the above cases.
Step 3: Analyze Statement 1 independently
(1) a < 1
Insufficient to answer
Step 4: Analyze Statement 2 independently
(2) b is a positive odd integer
Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
(1)From Statement 1, 1 < a < 1
(2) From Statement 2, b > 0
Following cases are possible:
As we do not have a unique answer, the combination of statements is insufficient to answer.
Answer: E
Z=2^{a}×5^{b}×7^{c}
A positive integer Z can be expressed in terms of its prime factors as above, where a, b and c are positive integers. Is 3^{a−b}<9?
(1) Z is divisible by 40 but not by 50
(2) Z^{a−b}=2^{6}×5^{2}×7^{4}
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Step 3: Analyze Statement 1 independently
(1) Z is divisible by 40 but not by 50
a – b ≥ 2
So, answer to the asked question is: NO
Statement 1 is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
Thus, the answer to the asked question is: NO
Statement 2 is sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at unique answers in Steps 3 and 4, this step is not required
Answer: Option D
What is the maximum possible power of 4 in the number that is obtained when the product of the first 15 positive integers is subtracted from the product of the first 20 positive integers?
Given
To Find: Maximum power of 4 that divides (20! – 15!)
Approach
2. To find the maximum power of 4 in 15!, we will first need to find the maximum power of 2 in 15!, as 4 = 2^{2}
3. We will use the same logic to find out the power of 2 in 15!
Working Out
For any integer z, the function PODD(z) is defined as the product of all odd integers between –z and z, inclusive. Which of the following numbers will not divide PODD (15) completely?
I. 3^{8}
II. 5^{5}
III. 7^{3}
Given:
To find: PODD(15) will not be divisible by which of the given 3 options?
Approach:
Working Out:
Finding the power of 5 in PODD(15) and evaluating Option II
Looking at the answer choices, we see that the correct answer is Option E
If x > 0 such that what is the value of
Given
Approach
Working Out
As x > 0, cannot be equal to 7. Hence, =7
Answer: D
If x and y are distinct positive integers, such that
What is the value of (x)^{(y+1)} ?
Given Info:
To Find:
Approach:
Working out:
On comparing both sides of the above equation as both are to the same base 2, we get
⇒x^{2}=5x−4
Subtracting 5x  4 from both sides we get
Sowehavex=1 or x=4
⇒ y+1=2
⇒ y=1
Case 2: x=4 (Value derived from Equation 1)
⇒ y + 4 = 5
⇒ y = 1
Pair1:(x,y)=(1,1)(FromCase1)
Pair2:(x,y)=(4,1)(FromCase2)
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