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If $p,q$ are distinct primes and $a$ is not divisible by $p$ or $q$, then $\gcd(a, pq)=1$.
I want to show this using linear combinations, so that a linear combination of $a$, and $py$ will give $1$. So for some $x,y,x',y'$:
$ax+py = 1 = ax'+qy'$, and
Not sure where to go from here. Hints appreciated.