Question

Prove that for any integer m, m^5 is congruent to m modulo 5.

Answer 1

Here's a direct proof: note that

n5−n=n(n4−1)=n(n−1)(n+1)(n2+1)

You need to prove that this number is divisible by 2 and 5.

For divisibility by 2, note that either n or n+1 must be even.

For divisibility by 5, you should prove that if neither n nor n±1 is divisible by 5 (i.e. if n≡2 or 3mod5) then n2+1 is divisible by 5