GCD domain and properties

Question

Let $A$ be a commutative GCD domain (not necessary UFD or Bezout) and $a,b,c$ elements of $A$ such that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$. Is it true that $\gcd(ab,c) = 1$ ?

Answer 1

One can prove a more general result: If $\gcd (a,b)=1$, then $\gcd (a,bc)=\gcd(a,c)$ for all $a,b,c$ in a GCD domain.

For the proof, just notice that if $d$ is a common divisor of $a$ and $bc$, then we also have $d\mid ac$, thus $d\mid\gcd(ac,bc)=c\gcd(a,b)=c$, so $d$ is also a common factor of $a$ and $c$.