Source: Official Guide for the GMAT 12th Ed. Section 5.3 Problem Solving; #142

1

# If y is the smallest positive integer

If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be 2, 5, 6, 7, 14

### 2 Explanations

1

ANNA CHENGUELLY

Hello,
Could you please kindly advise cause I thought that a perfect square has an odd number of factors. Thank you.

Apr 24, 2017 • Comment

Cydney Seigerman, Magoosh Tutor

Happy to help :)

You're correct that a perfect square has an odd number of total factors. Here, however, we're only considering the prime factors. Conversely, as Mike mentions, the prime factors in perfect square will appear as pairs, which gives us an even number of prime factors in the prime factorization of the square. Let's consider 16 to see this difference:

Factors of 16: 1, 2, 4, 8, 16 (5 factors)
Prime factorization of 16: 2*2*2*2 = 2^4 (even number of prime factors)

The prime factorization is composed of two pairs of 2, giving us an even number of prime factors. On the other hand, we see that 16 has a total of 5 different factors. :) And this reflects the descriptions above!