Let $m = 2p$
If p is an odd prime, prove that $a^{2p - 1} \equiv a \pmod {2p} \iff a^{m - 1} \equiv a \pmod m$.
I have no idea on how to start. I was trying to find a form such that
$a^{m - 2} \equiv 1 \pmod m$. But I got stuck. Can someone give me a hint here?