Something similar to Euler's theorem

Question

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$.

Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds

$a^{\frac{\varphi(n)}{d}} \equiv 1 \mod n$ ?

Answer 1

More generally let $G_1,G_2$ be finite groups of orders $n,m$ (in your case, $(\mathbb{Z}/p)^\times$ and $(\mathbb{Z}/q)^\times$) and $d := gcd(n,m)$. Then for any element $a \in G_1 \times G_2$ we have $a^{nm/d}=1$. In fact, $a=(a_1,a_2)$ with $a_i \in G_i$, and since $n \cdot m/d$ is a multiple of $n$ and $a_1^n=1$, we have $a_1^{nm/d}=1$. The same argument shows $a_2^{nm/d}=1$.