Is it true that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $\gcd(ac,b) = 1$? 
Is it true that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $\gcd(ac,b) = 1$?

I know that $\gcd(a,b) = 1$ means that there exist integers $m$ and $n$ such that
$am + bn = 1$
Same thing for $\gcd(a,c)$: we have $ax + cy = 1$.
But how can I relate this to $\gcd(ac,b)$?

Let $a=2, b=c=3$. Then $\gcd(a,b)=\gcd(a,c)=1$ and $\gcd(ac,b)=3$.
You can prove that $\gcd(a,bc)=1$, however.
