Let $p$ be a prime integer, and let $m,k \in \mathbb{N}$. If $m \equiv k$ (mod $(p-1)$), show that:
$u^k \equiv u^m$ (mod $p$), $\forall u \in \mathbb{Z}$
If $u \equiv v$ (mod $p$), then $u^k \equiv v^m$ (mod $p$), $\forall u,v \in \mathbb{Z}$
Couldn't find anything related with this. The only thing related with $(p-1)$ I found was Wilson's Theorem, but I can't see any relation with that question. Can anyone show how can I start to solve that? Thanks.
