Euler’s Formula application for distinct odd primes with gcd 1

Question

I am looking at the following question from my undergraduate Number Theory textbook:

Show that if p,q are different odd primes, and if gcd(p,q)=1, then a^$\Phi$(pq)/2 $\equiv$ $1$ mod $pq$.

So far, the approach I have taken is trying to split up a^$\Phi$(pq)/2 = (a^$\Phi$(p)/2)^$\Phi$(q) and apply Euler's theorem but I don't think I am really getting anywhere.

Any help would be greatly appreciated, thank you!

Answer 1

Note: If $\gcd(a,pq) \neq 1$, then clearly $a^{ \phi (pq) / 2 } \neq 1 \pmod{pq}$. So, I'm assuming that $\gcd(a,pq) = 1$.

Hint: $\phi (pq) = (p-1)(q-1)$, where both of the terms on the right are even, since they are odd primes.

Hence $a^{\frac{(p-1)(q-1) } {2} } \equiv \left( a^{p-1} \right) ^ { \frac{q-1}{2} } \equiv 1 \pmod{p}$.
Similarly for $\pmod{q}$, and hence for $\pmod{pq}$ (since they are distinct primes).