Let p is prime number.
Prove . How to do it?
It is clear that:
, ,
Is it and why?
Thank you.
Since $a^p\equiv a\,(mod\, p)$, $a=b+qp$ for some q. Thus
$$a^p-b^p=\sum_{k=1}^p{{p}\choose{k}}b^{p-k}(qp)^k=pb^{p-1}qp+\sum_{k=2}^p{{p}\choose{k}}b^{p-k}(qp)^k$$
We're done.
Worth remembering: $p$ divides each binomial coefficient ${{p}\choose{k}}\text{ for }1\leq k<p$.