Problem - 4171
Let $p$ and $q$ be two distinct primes, and integer $a$ is co-prime to both $p$ and $q$, show $$a^{(p-1)(q-1)}\equiv 1\pmod{pq}$$
It is clear that $a$ is co-prime to $pq$ and $\varphi(pq)=(p-1)(q-1)$. Therefore by Euler's theorem, we have $$a^{(p-1)(q-1)}\equiv 1\pmod{pq}$$