Showing that $(a,b)=1$ where $d=(m,n)$ and $m=ad$ and $n=bd$ If $d=(m,n)$, then $d|m$ and $d|n$ so there must exist integers $a$ and $b$ such that $m=ad$ and $n=bd$. Now $d=(m,n)=(ad,bd)=d(a,b)$ and so $(a,b)=1$. Is this correct?

Yes, the proof is flawless.You have proved that$$ d=(m,n)=(ad,bd)=d(a,b)\implies (a,b)=1$$
