GMAT Problem Solving- If x and y Are Positive Odd Integers, And Both Numbers Have An Odd Number Of Positive Divisors

Question: If x and y are positive odd integers, and both numbers have an odd number of positive divisors, which of the following could be the value of x-y?

4818
5174
5320
5482
5566

“If x and y are positive odd integers, and both numbers have an odd number of positive divisors,”- 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 number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

The given condition states that x and y are both positive odd integers. The case further identifies that if x and y are both positive odd integers and they have the same positive odd divisors, then the value of x-y needs to be evaluated.
In relevance to even and odd positive integers, there are certain key concepts that should be known.
Most integers have an even number of positive divisors.
For example, 12 has 6 divisors, 35 has 4 divisors, and 13 has 2 divisors.
However, squares of integers always have an odd number of divisors.
For example, 25 has 3 divisors, 36 has 9 divisors, and 81 has 5 divisors.
Accordingly, if x and y both have an odd number of positive divisors, then x and y squares of integers.
So, let’s let x =\(j^2\) and let y = \(k^2\) for integers j and k
It is important to find a possible value of x – y
This implies we can write: x – y = \(j^2\)- \(k^2\)
The factor which one is supposed to get: (j + k)(j – k)
Since j and k are both odd, we know that (j + k)(j – k) = (odd + odd) (odd - odd) = (even)(even) = (2 times some integer)(2 times some integer) = (4)(some integer)
Another key concept important in this aspect is that If integer N is divisible by 4, then the number created by the last 2 digits of N must be divisible by 4.
Hence, amongst all the options given for this case, the one which is divisible by 4 is 5320, considering its last 2 digits 20 which is divisible by 4. Therefore the value of x-y equals 4

Correct Answer: C

Approach Solution 2:

The given case is that both x and y are odd positive integers and both have odd positive divisors.
To find the value of x-y, the following equation needs to be considered:
x = \((2a+1)^2\), which is an integer
Y = \((2b+1)^2\), which is an integer
x - y = 4\(a^2\) + 4a + 1 - 4\(a^2\) - 4b - 1 = 4(\(a^2\) + a - \(b^2\)- b)
From the above equation, it can hence be derived that the number is divisible by 4 and the only number from the given 5 options is 5320.

Correct Answer: C

Approach Solution 3:

The case in question is when both x and y have odd positive divisors and are both odd positive integers.
The following equation must be taken into account in order to get the value of x-y:

x = \((2a+1)^2\)an integer
Y = \((2b+1)^2\)an integer
x - y = 4\(a^2\) + 4a + 1 - 4\(a^2\) - 4b - 1 = 4(\(a^2\) + a - \(b^2\)- b)

The only number from the supplied five alternatives that can be divided by 4, according to the above equation, is 5320.

Correct Answer: C

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GMAT Problem Solving- If x and y Are Positive Odd Integers, And Both Numbers Have An Odd Number Of Positive Divisors

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