Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.

Question

I was told to prove $\gcd(ab,c)=1$ then $\gcd(a,c)=1$.

I picked a number $p$ that goes into $ab$ and $c$, so $ab=px$ and $c=py$. but now what?? I tried $abc=p^2xy$ but then I can't.

Please help me!

Answer 1

Hint $\,\ d\mid a,c\,\Rightarrow\,d\mid ab,c,\,$ but $\,ab,c\,$ are coprime so $\,\ldots$

Answer 2

$\gcd(ab,c)=1 \Leftrightarrow \exists s,t$ such that $(ab)t+cs=1$. But then you have that $a(bt)+cs=1$ which give you $\gcd(a,c)=1$.

You only use that for $m,n\in\mathbb{Z}$, $\gcd(m,n)=1$ if-f $\exists s,t\in\mathbb{Z}$ such that $mt+ns=1$.