Let $a = p_1\cdots p_n$ and $b = q_1\cdots q_m$ be two non-zero elements of a unique factorization domain $R$ and their respective factorization in irreducible factors, such that no $p_i$ is associated with a $q_j$ (i.e., $\nexists k\in R^*:p_i=kq_j$). Let $GCD(a,b)$ be the set of all greatest common divisors of $a$ and $b$ in $R$. Then I want to conclude that $1_R\in GCD(a,b)$
Once $1_R$ divides all elements of $R$, it's equivalent to show that if $d$ is a common divisor of $a$ and $b$, then $d$ divides $1_R$, which is equivalent to $d$ be an unity element, or that $d$ is associated with $1_R$. In any if these ways I don't see how to use the hypothesis that $a$ and $b$ don't have associated irreducible factors. Can someone give me a hint on this?