Gcd and lcm of $a_1, a_2, \dots,a_n$ exist in $R$ when $R$ is a UFD.

By existence and uniqueness of prime factorizations, divisibility reduces to divisibility in each prime component, i.e. $\ p^{\large a} q^{\large a'}\!\cdots\mid p^{\large b} q^{\large b'}\!\cdots\!\iff p^{\large a}\mid p^{\large b},\,\ q^{\large a'}\!\mid q^{\large b'},\,\ldots\ $

So $\,\ a\mid b,c\iff p^{\large a}\mid p^{\large b},p^{\large c},\ \ q^{\large a'}\mid q^{\large b'},q^{\large c'},\ldots\ $ each which obey

$$\quad\ \ \ \ \ \ p^{\large a}\mid p^{\large b},p^{\large c}\! \iff a\le b,c \iff a\le \min\{b,c\} \iff p^{\large a}\mid p^{\large \min\{b,c\}}$$

$\!\begin{align}{\rm So}\ \ a\mid b,c&\iff p^{\large a}\mid p^{\large \min\{b,c\}},\ q^{\large a'}\mid q^{\large \min\{b',c'\}}\ldots\\[.5em] &\iff a\mid \color{#c00}{p^{\large \min\{b,c\}} q^{\large \min\{b',c'\}}}\ldots \end{align}$

Remark $\ $ Above we employ the universal characterization / definition of gcd and lcm, i.e.

$$\begin{align} a\mid b,c\iff a\mid \color{#c00}{\gcd(b,c)}\\ b,c\mid a\iff {\rm lcm}(b,c)\mid a\end{align}\qquad\qquad$$

The $ $ lcm $ $ case is just the dual $\, \ b,c\mid a\iff \cdots,\ $ by reversing divisiblities in the gcd proof.