Let $R$ be a unique factorization domain (UFD). Let $a,b,c\in R$.
Prove that $\text{gcd}(ca,cb)=c \times\text{gcd}(a,b)$.
Proof: We know that since R-UFD then R-GCD domain. Suppose that $d=\text{gcd}(a,b)$ then we need to prove that $cd=\text{gcd}(ca,cb)$.
1) It is easy to see that $cd|ca$ and $cd|cb$.
2) We need to show that if $x|ca$ and $x|cb$ then $x|cd$. But unfortunately I am not able prove that.
I guess that here we need to use that $d=\text{gcd}(a,b)$.
Would be very grateful if anyone show to prove that 2) holds.
