Number of divisors?

Hmm, the "similar questions" presented by Yahoo seem to indicate this one might be simple for some of you. Ah well, here goes: Let n be a positive integer such that 5n has 36 positive divisors and 3n has 40 positive divisors. How many positive divisors can 15n have? How many values of n are... show more Hmm, the "similar questions" presented by Yahoo seem to indicate this one might be simple for some of you. Ah well, here goes:

Let n be a positive integer such that 5n has 36 positive divisors and 3n has 40 positive divisors.

How many positive divisors can 15n have?
How many values of n are possible?

[I have a solution which I believe is correct, but it's far from elegant; as such, whoever gets the most elegant solution will get best answer.]
2 answers 2