Given a group is abelian, o(x) and o(y) are finite, show o(x*y) is finite.

Does anyone have any ideas on this one that could help get me started?

Here is about as far as I've come so far.

we know:

there are integers m, n such that x^m = e and y^n = e

x*y = y*x

we want to show:

there is an integer j where (x*y)^j = e

the question also wants me to show that that o(xy) divides o(x)*o(y). I'm thinking that the fact that i'm going to be able to show:

there is an integer a where j * a = m * n

is going to be helpful with the first question, but i haven't figured out how yet