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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:

equation

Then, the statement of the problem requires that:

equation

and that:

equation

Solving this system of equations by substitution, we find:

equation

equation

equation

equation

equation

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

equation

which means that N has to be a perfect square.