elementary number theory - Suppose $gcd(a,b)=1$ and $c|ab$. Prove there exist integers $r$ and $s$ such that $c=rs, r|a, s|b$, and $gcd(r,s)=1$

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.

Answer

Say $g=\gcd(a,c)$ then $c=gd$ and $a=gx$ (so $d$ and $x$ are relatively prime). So we get

$$gd\mid gxb$$ and thus $d\mid xb$. But $x$ and $d$ are relatively prime so $d\mid b$ and thus $b=de$ for some $e$.

So put $r=g$ and $s=d$ and you are done.