Let $m,q,r,s$ be integers.
Suppose GCD($r$,$s$) = 1.
Suppose $m|qr$ and $m|qs$.
Show that $m$ divides $q$.
This was a result that was assumed in class. I couldn't see the reasoning behind it though.. please provide some guidance/hints.
What I've tried so far:
I note that since $(r,s)=1$, then either $m$ does not divide $r$, or $m$ does not divide $s$.
I suppose that $m$ does not divide $r$.
Then I note that $m|qr$ but m does not divide $r$.
I try to come up with a contradiction by assuming that $m$ does not divide $q$, but since $m$ isn't prime, this doesn't get me far.
Am I going down the wrong path? Is there something I'm failing to consider?