Problem - 4170
Let $p$ be a prime and integer $a$ is co-prime to $p$, show that $$a^{p(p-1)}\equiv 1\pmod{p^2}$$
It is clear that $a$ will be co-prime to $p^2$. Meanwhile, we have $\varphi(p^2)=p(p-1)$. Therefore, by Euler's theorem, we have $a^{p(p-1)}\equiv 1\pmod{p^2}$.