Let $D$ be a unique factorization domain, and let $a,b,c\in D\setminus\{0\}$
Prove that:
(i) If $1_D$ is greatest common divisor of $a,b$ and of $a,c$, then $1_D$ is a greatest common divisor of $a,bc$
(ii) If $a\mid bc$ and $1_D$ is a greatest common divisor of $a,b$, so $a\mid c$
(iii) Are the above statements true if $1_D$ is replaced by $u∈U(D)$?
Thank you!!