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.


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