If x and y are Positive Integers, What is the Value of x - y? GMAT Data Sufficiency

Question: If x and y are positive integers, what is the value of x–y?

(1) The greatest common divisor of x and y is 1
(2) 13x/y is a positive integer with exactly two positive divisors

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D
Solution and Explanation:
Approach Solution 1:

Given in the question that x and y are positive integers. It has asked to find out the value of x- y.

1) GCD of x and y equals one.
Any pair of two prime numbers we choose will have a gcd of 1, but their differences won't be the same.
2,3 Diff = 1
2,5 Diff = 3
2,13 Diff = 11
It's not enough

2) 13x/y has precisely two positive divisors and is a positive integer.
Due to the fact that only prime numbers have precisely two positive divisors, 13x/y is a prime number.
If y = 13 and x is any prime number, then 13x/y can be prime. However, if x = 2, y = 13, Diff = 11, and if x = 5, y = 13, Diff = a it is not enough.
13x/y is a prime integer, and both of its combined GCDs, x=2, y=13 = 2, y = 13, Diff = 11, and x = 5, y = 13, Diff = 8, are 1.
It's not enough
Response: E is the correct answer.

Approach Solution 2:

Given the question, we have positive integers x and y.

What does x-y equal in terms of value?

Statement 1: One is x and y's biggest common factor.

Statement 1 is true for a variety of x and y values. These two are
Case A: If x and y are both 1, then 1 is their greatest common factor. In this instance, x - y = 1 - 1 = 0 is the response to the target question.
Case b: The greatest common factor of x and y is 1 if x = 1 and y = 2. In this instance, x - y = 1 - 2 = -1 is the response to the goal question.
Statement 1 is insufficient because we lack certainty regarding the goal question.

2. 13x/y has exactly two positive divisors and is a positive integer.

Also known as 13x/y, this number is a prime.
Statement 2 is true for a variety of x and y values. These two are
Case A: 13x/y = (13)(1)/(1) = 13, which is a prime number, if x and y are both 1. In this instance, x - y = 1 - 1 = 0 is the response to the target question.
Case b: If x is 5 and y is 13, then 13x/y = (13)(5)/(13) = 5 and this number is a prime. In this instance, x - y = 5 - 13 = -8 is the response to the goal question.
Statement 2 is insufficient because we lack certainty regarding the goal question.

Statements 1 and 2 taken together
There are numerous x and y variables that are true for BOTH claims. These two are
Case A: X and Y are both 1. In this instance, x - y = 1 - 1 = 0 is the response to the target question.
Case B: With x = 5 and y = 13,
In this instance, x - y = 5 - 13 = -8 is the response to the goal question.

E is the correct answer.

Approach Solution 3:

Given in the question that x and y are positive integers. It has asked to find out the value of x- y.

Case 1:
The GCD of two numbers is 1.
There can be many numbers whose GCD is 1 but the difference can change.
This information is not enough to get the answer.

Case 2: This says that 13x/y has exactly two divisors.
This means 13x/y is a prime number.
It has two divisors - the first is 1 and the second is 13x/y itself.
Taking these two values, they will give different results each time. Hence this statement is not sufficient to get the answer.
Together also they are not sufficient to find the answer.

Hence E is the right choice.

“If x and y are positive integers, what is the value of x–y?” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”.

To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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