The GCD of Certain Multiples of Coprimes

Question

Let $m,n\in \mathbb{Z}$, such that $\gcd(m,n)=1$, and let $p,q\in \mathbb{Z}$. Is there anything that we can conclude about $\gcd(pn,qnm)$? I am asking this because it might help me answer the truth of the statement: if $\gamma | \gcd(pn,qnm)$ (with $n,m$ coprime), then $\gamma | pqn$.

Answer 1

If $\gamma | \gcd(pn ,qnm)$ , in particular it is a common divisor of $pn$ and $qnm$. Hence $$\gamma | pn \ \Longrightarrow \ \gamma |pqn$$