# Divisibility of the difference of powers

Consider the following theorem:

For any $a, b \in \mathbb{Z}^+$, there exist $m, n \in \mathbb{Z}$ such that $m > n$ and $a\ |\ b^m - b^n$.

What's the best way to prove it? I have an idea (and I know it's true because of that idea), but I don't know how rigorous it is to constitute a proof.