Let $a∈\Bbb Z$ and let $p$ be a prime number.
Prove that $$p|a{^p}+(p−1)!a\quad\text{and}\quad p|(p−1)!a{^p}+a.$$
I was thinking of using Fermat's Little Theorem to solve this question.
From the theorem I know that
$$a^{(p-1)}\equiv1\pmod p.$$
Also I was wondering if I could use Wilsons Theorem.
From Wilson I can show that $$a^{(p-1)} (p-1)!\equiv(p-1)! \pmod p$$
is equivalent to $$-a^{(p-1)}\equiv -1 \pmod p.$$
However, I feel like I am going in a circle. Can anyone give me a hand.
Your thoughts are correct.
By Fermat’s Little Theorem for any integer $a$, $$a^p\equiv a\pmod{p}$$ and by Wilson’s Theorem $$(p-1)!\equiv -1\pmod{p}.$$ The result then follows from the multiplicative and reflexive properties of congruences.