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 $\gamma | pqn$.
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1All we can say is that $$\gcd (pn , qnm) = n \gcd (p , qm)$$ – Crostul Apr 05 '17 at 08:01
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I see $n$ is common in $pn$ and $qnm$ , so $n$ is $\gcd$ right?? – Fawad Apr 05 '17 at 08:02
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@Fawad $n$ is a common divisor, but it might not be the largest one. – Crostul Apr 05 '17 at 08:04
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If $\gamma | \gcd(pn ,qnm)$ , in particular it is a common divisor of $pn$ and $qnm$. Hence $$\gamma | pn \ \Longrightarrow \ \gamma |pqn$$
Crostul
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