OG - Quantitative: #169

IF n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is?

a) 6
b) 12
c) 24
d) 36
e) 48

The answer is B.

I must be missing something. If n = 48, 48^2 is divisible by 72. (48*48 = 2304; 2304 / 72 = 32). The largest integer that divides 48 is 48.

I must be tired, so can someone please explain the hole in my logic?

Thanks!

This is one of those 700-level problems that must be read extremely carefully in order to select the correct answer. The key term in this problem is "must," since it confines us to an answer that undoubtedly divides n. This is very different from a problem that says "could," since a problem with this phrasing is open to any answer choice that happens to work.

In our problem, if n^2 is divisible by 72, then n^2 MUST contain the prime factors 2,2,2,3,3. N^2 COULD contain other factors, but those listed are definite.

Next, if n^2, or (n)(n), contains the prime factors 2,2,2,3,3, then those factors MUST be divided equally among each n (as one n must be identical to the other). So, splitting up the prime factors 2,2,2,3,3 among each n, we see that each n MUST contain the factors 2,3, and the trap answer is (A).

A is not correct, though, since we are still left with one extra 2 as a factor. We know that this 2 is included in n^2, so it must be contained by one of the n's, leaving us with one n = 2,2,3 and the other n = 2,3. However, since n must be identical to n, and one n MUST contain 2,2,3, the other n MUST contain the same factors 2,2,3 as well.

Since n MUST contain at least the factors 2,2,3 we can be certain that n is a multiple of 12. Therefore, 12 is the largest positive integer that MUST divide n.

The correct answer is B

-dan