2
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.