If $p,q$ are distinct primes and $a$ is not divisible by $p$ or $q$, then $\gcd(a, pq)=1$

If $a$ is not divisible by $p$ or $q$ then indeed there exist integers $x$, $x'$, $y$, and $y'$ such that $$ax+py=1\qquad\text{ and }\qquad ax'+qy'=1.$$ Now isolate $py$ from the first and $qy'$ from the second equation, and multiply the two results together. Can you finish from here?

$ax+py = 1$ and $ ax'+qy'=1$. Rearranging, we have $py = 1-ax$ and $qy'=1-ax'$. Multiplying, we get $$pyqy'=(1-ax)(1-ax')=1-a (x+x')+a^2xx'.$$

Hence $$pq(yy')+a (x+x'-axx')=1. $$