This is a typical pigeonhole principle problem. There is a trick to making work. And you're going to feel really stupid until you see the trick. Accept that. If you keep pounding on this and aren't making progress, put it down for a while (perhaps overnight) so you can see it from a fresh perspective.
Suppose you are working in \(\mathbb{Z}_{11}\) and you compute \([5]^m\) for various values of m.
Perhaps you are thinking of \(13^m-13^n\) as a single algebraic expression, and then trying to figure out when it might be divisible by 13. However, remember that "x-y is divisible by k" is equivalent to \(x \equiv y \pmod{k}\). Maybe the problem is easier to think about if you think about \(13^m\) and \(13^n\) as separate numbers, and then try to compare them in some modular way.