Show that if $p$ is a odd prime number, then $p\mid2^{p-1} + (p -1)!$ [duplicate]

If p is prime, prove that p divides $a^p +(p-1)!a$ and p divides $(p-1)!a^p +a$

The answer is related to Fermat's Little theorem, but I can't figure out how to incorporate a factorial into the theorem. Any help would be much appreciated.


Solution 1:

Hints:

-- FLT: For any $\;a\in\Bbb Z\;,\;\;a^p\equiv a\pmod p\;$

-- Wilson's Theorem: for any prime $\;p\in\Bbb N\;,\;\;(p-1)!\equiv -1\pmod p\;$