Showing that $(ab,d^2)=1$ where $d=(m,n)$ and $m=ad$ and $n=bd$

Question

If $d=(m,n)$ then $m=ad$ and $n=bd$. How can I show that $(ab,d^2)=1$?

Answer 1

It's not true. Take $a=2$, $b=3$, $d=6$. Then $m=12$, $n=18$, so that $(n,m)=6=d$, yet $(ab,d^2)=6\neq 1$.

I hope this demonstrates the importance of trying small values first before proving something. Not only does it give you feel for the problem, potential proofs, and in some cases solutions; it also prevents you from trying to solve theorems that aren't even true.