Question

Find all positive integers n such that n^4-1 is divisible by 5.

Answer 1

Fermat's little theorem: If a is not divisible by p, Fermat's little theorem is equivalent to the statement that $a^{p-1}-1$ is an integer multiple of p, or in symbols:

$a^{p-1}\equiv 1 \mod p$

so, $p=5, \,a=n$ and $n=1,2,3\, \text{or}\, 4 \mod 5$