Show that $\gcd(an,bn)=|n|\gcd(a,b)$.
Question: Let $k=\gcd(a,b)$. Then $kq_1=a,kq_2=b$. Then $kq_1n=an,kq_2n=bn$. Hence, $kn|an$ and $kn|bn$. I'm having difficulty showing that, however, it's necessarily true that $kn=\gcd(an,bn)$.
I attempted proof by contradiction, i.e., suppose that there is an $r$ such that $r>kn$ and $r=\gcd(an,bn)$, but that gave me the trouble of showing that $r/n\in\mathbb Z$.
Note: One other similar question was submitted, but this is not the same. I want detailed clarification on the last bit.