Prime Factorization is Unique, Correct?

[darthbob88]

As an extra credit question, my professor asked us to determine whether two functions are one-to-one, whether there exist two different sets of inputs that produce the same output. The functions map positive integers for X to positive integers for Y; F(m, n) = 3^m * 5^n, and G(m) = 3^m * 6^n.

Obviously, so long as m and n are restricted to positive integers, the first is merely the prime factorization of whatever the output is, and must be one-to-one, since there is no possible way to rewrite 3^m as 5^n, where m and n are not 0.

The second makes me a little dubious; admittedly, there is no mathematical reason why it should not be one-to-one, for the same reasoning as above. Still, the fact that 6^n can be re-written as 2^n * 3^n leads me to believe that this function is not entirely one-to-one.

Quite frankly, am I or am I not being paranoid about this?

[Kuroneko]

Prime factorization of integers greater than 1 is unique up to order. As for your functions, both of them are one-to-one. Hint: given an arbitrary output of G(m,n), first show that one there can be at most one n that gives that output; from this, the rest (uniqueness of pair (m,n)) is easy. Starting with n is important.
[Edit: minor clarification (on second thought, let's be more vague so as to not give it away).]