Brent@GMATPrepNow wrote:How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve

Source: GMAT Prep Now
Difficulty level: 650

Answer: C

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IMPORTANT CONCEPT: The prime factorization of a perfect square will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even numbers allow us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2X2X2X2X2X2X3X3 = (2X2X2X3)(2X2X2X3) = (2X2X2X3)²

--now onto the question-----------------------------------------------------------------------------

12,500 = 2x2x5x5x5x5x5 = (2x2)(5x5)(5x5)(5)
Since we need an even number of each prime [in order for the product to be a perfect square], we need only determine how many different perfect squares can be achieved by using various configurations of (2x2), (5x5) and (5x5)

Let's list them:
1) (2x2) = 4
2) (2x2)(5x5) = 100
3) (2x2)(5x5)(5x5) = 2500
4) (5x5) = 25
5) (5x5)(5x5) = 625
6) 1 [a factor of all positive integers]

So, there are 6 factors of 12,500 that are squares of integers
Answer: C