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

Question

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.

Answer 1

Hints:

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

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