Let $D$ be an integral domain with an Euclidean valuation. Let $a,b \in D$ not both 0. Let $(a,b) =\{ax +by \vert x,y \in D\}$ . $(a,b)$ is an ideal in $D \ne \{0\}$ . We choose $d \in (a,b)$ such that $\mu(d)$ is the smallest positive value. Then $(a,b)$ is generated by $d$ and has the property $d \vert a$ and $d \vert b$ . Two elements $x$ and $y$ in $D$ are associate if and only if $\mu(x) =\mu(y)$ . So $d$ is unique up to a unit in $D$ . Hence $d$ is the greatest common divisor of $a$ and $b$ .