Greatest common divisor

Let $R$ be an integral domain and let $a,b\in R$. A greatest common divisor of $a$ and $b$ is an element $d\in R$ such that:

1. $d\mid a$ and $d\mid b$, and
2. if $c\mid a$ and $c\mid b$, then $c\mid d$.

A gcd is unique up to associates (so one often fixes a “normal form” when possible). When gcds exist for all pairs, one can define lcms and obtain identities relating gcd and lcm .

Examples:

• In $\mathbb{Z}$, $\gcd(12,18)=6$.
• In $\mathbb{Q}[x]$, a gcd of $x^2-1$ and $x^2-3x+2$ is $x-1$ (up to nonzero rational scalars).
• For any $a\in R$, a gcd of $a$ and $0$ is $a$ (up to associates).