I'm trying to prove that if $a|c$ and $b|d$ and $\gcd(c,d)=1$ then $\gcd(a,b)=1$
So far, I have assumed that:
Since $\gcd(c,d) = 1$ then by EEA, $$\gcd(c,d) = 1 = cx + dy$$ for some $x,y$ that are integers. And since $a|c$ and $b|d$ then $c=am$ and $d=br$ for some $m,r$ that are integers.
I just don't know where to go from here. Thanks you.