I have the following problem on greatest common divisors:
Suppose $\gcd(a,b)=1$ and $c|ab$. Prove there exist integers $r$ and $s$ such that $c=rs, r|a, s|b$, and $\gcd(r,s)=1$.
Attempt: Basically, up to this point I've tried a whole lot of algebraic manipulation. Since $\gcd(a,b)=1$ we have $1=ma+nb$, for some integers $m,n$. Also, $ab=kc$ for some integer $k$. We aim to show (to begin with) that $r|a$, that is, that $a = rp$, for some integer $p$. So basically, I've started with the equation $1=ma+nb$ and multiplied by $a, b, ab$ etc., and I just can't seem to be able to express $a$ in the form above. I then looked at some of the results on prime factorisation, but they don't seem to shed any light on the situation.
Any help or hints would be appreciated.