Prove that if $p$ is a prime number, then $p|n^p − n$ for $n ≥ 1$
I tried proof by induction for this but was not able to even prove the base case (I tried it for $n=1$ since we're trying to prove for $n≥1$). I'm more used to seeing these kinds of problems with actual numbers instead of the abstract prime $p$. So what I'm understanding for this problem, it's asking me that any prime $p$ should be able to divide $n^p - n$, but isn't this false? If we let $n = 1$ and $p = 2, 2|1-1$ shouldn't be right. Can someone elaborate for me or give a few hints? I think proof by induction should be the right direction though...