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This problem trains for: SAT-I, AMC-8, GMAT.

If an integer N has 36 distinct positive divisors and its square has 119 distinct positive divisors, what is the largest exponent present in the prime factorization of N?

Let us figure out first what possible factorizations N could have. If 36 is obtained from the product:

equation

then a possible factorization of N is:

equation

the square of N has the factorization:

equation

and the number of divisors of the square is:

equation

Similarly, for a different possible product:

equation

equation

equation

equation

And again, with a different choice of factors:

equation

equation

equation

equation

which happens to be the correct answer. We can stop here, although we know there are a couple more possibilities left to explore.

Think: can there be two different such options that lead to the same number of divisors for the square?