Let p and q be any two distinct prime numbers and define the relation $aRb$ on integers $a,b$ by: $aRb$ iff $b-a$ is divisible by both $p$ and $q$. For this relation $R$: $\\$
(1) Prove that $R$ is an equivalence relation. $\\$
(2) Show that the equivalence classes of $R$ correspond to the elements of $\mathbb{Z}_{pq}$. That is: $[a] = [b]$ as equivalence classes of $R$ if and only if $[a] = [b]$ as elements of $\mathbb{Z}_{pq}$. Check for yourself that (1) does hold still, but (2) need not be true if $p, q$ are not prime. $\\$
You may use the following lemma: $\\$
If p is prime and $p\mid mn$, then $p \mid m$ or $p \mid n$.
Indicate in your proof the step(s) for which you invoke this lemma. $\\$
I proved part (1), but I didn't use the lemma yet, so I'm pretty sure I need to use it in part (2), but I'm not sure where to start.