How to show $a,b$ coprime to $n\Rightarrow ab$ coprime to $n$? Let $a,b,n$ be integers such that  $\gcd(a,n)=\gcd(b,n)=1$. How to show that $\gcd(ab,n)=1$? 
In other words, how to show that if two integers $a$ and $b$ each have no non-trivial common divisor with and integer $n$, then their product does no have a non-trivial common divisor with $n$ either.   
This is a problem that is an exercise in my course. 
Intuitively it seems plausible and it is easy to check in specific cases but how to  give an actual proof is not obvious.  

Let $P(x)$ be the set of primes that divide $x$. Then $\gcd(a,n)=1$ iff $P(a)$ and $P(n)$ are disjoint. Since $P(ab)=P(a)\cup P(b)$ (*), $\gcd(a,n)=\gcd(b,n)=1$ implies that $P(ab)$ and $P(n)$ are disjoint, which means that $\gcd(ab,n)=1$.
(*) Here we use that if a prime divides $ab$ then it divides $a$ or $b$.
