Basically, this is the same question as Why is the last digit of $n^5$ equal to the last digit of $n$?
What I want to prove is
$n^5 ≡ n$ mod 10
Since I'm studying Euler's Phi Function, I know that the proof of this is related to it. So I'm looking to prove this using the Phi function. A comment on the original question suggests $φ(10)=4$ but I don't see how I can use this. Anyone can point me in the right direction? Thanks