The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This problem trains for: AMC-8, AMC-10, AMC-12.

If the number of divisors of N is three times larger than the number of divisors of M and M has 2 divisors less than N, which of the following statements must be true?

Denote the number of divisors of N and M with:

Then, the statement of the problem requires that:

and that:

Solving this system of equations by substitution, we find:

Therefore, the prime factorization of N can only be of the form:

which means that N has to be a perfect square.