elementary number theory - Let $p$ be a prime. Prove that $p$ divides $ab^p−ba^p$ for all integers $a$ and $b$.

Let $p$ be a prime. Prove that $p$ divides $ab^p−ba^p$ for all integers
$a$ and $b$.

Answer

$$ab^p-ba^p = ab(b^{p-1}-a^{p-1})$$

If $p|ab$, then $p|(ab^p-ba^p)$ and also if $p \nmid ab$, then gcd$(p,a)=$gcd$(p,b)=1, \Rightarrow b^{p-1} \equiv a^{p-1} \equiv 1\pmod{p}$ (by Fermat's little theorem).

This further implies that $\displaystyle{p|(b^{p-1}-a^{p-1}) \Rightarrow p|(ab^p-ba^p)}$.

Q.E.D.