Challenge question: If x and y are positive odd integers, a

Brent@GMATPrepNow wrote: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?

A) 4818
B) 5174
C) 5320
D) 5482
E) 5566

Source: www.gmatprepnow.com
Difficulty level: 700+
Answer: C

Key concept: 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.

So, if and x and y both have an odd number of positive divisors, then x and y squares of integers.
So, let’s let x = j² and let y = k² for integers j and k

NOTE: If x is odd, then j must be odd, and if y is odd, then k must be odd.

Our goal is to find a possible value of x – y
We can write: x – y = j² - k²
Factor 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)

ASIDE: If an integer is even, we can rewrite that number as 2 times some number. For example, we can take 10 and rewrite it as (2)(5)

So, it must be the case that the correct answer must be divisible by 4

Key concept: If integer N is divisible by 4, then the number created by the last 2 digits of N must be divisible by 4.

Check the answer choices.
A) 4818. 18 is NOT divisible by 4. Eliminate A.
B) 5174. 74 is NOT divisible by 4. Eliminate B.
C) 5320. 20 IS divisible by 4. Keep.
D) 5482. 82 is NOT divisible by 4. Eliminate D.
E) 5566. 66 is NOT divisible by 4. Eliminate E.

Answer: C

Cheers,
Brent