Let a prime number $p$ and a natural number $m$. Prove that $m^p-m$ is divisible by $p$.
We were given a hint to use a multinomial coefficient. But I'm not really sure how it helps me. If I say that $m=1+1+...+1$ I get : $\sum_{k_{1},k_{2},\ldots,k_{m}}^{p}{p \choose k_{1},k_{2},\ldots,k_{m}}$ I remember we proved something in class abount $p \choose k$ being divisible by $p$ only if $1\leq k < p$ but I'm not really sure how it helps me here...