Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$ Suppose $n$ is an integer which is not divisible by $5$. Prove that $n^4 \equiv 1 \pmod{5}$.

Like Timbuc,
$$n^4-1=(n-1)(n+1)(n^2+1)=(n-1)(n+1)(n^2-4+5)$$
$$=\underbrace{(n-2)(n-1)(n+1)(n+2)}+5(n-1)(n+1)$$
As $n$ must divide exactly one of any five consecutive integers and $5\nmid n,(5,n)=1,$
$n$ must divide one of the multiplicand underbrace
