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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:
Then, the statement of the problem requires 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.