Category:Definitions/Greatest Common Divisor

This category contains definitions related to Greatest Common Divisor.
Related results can be found in Category:Greatest Common Divisor.

Let $a, b \in \Z: a \ne 0 \lor b \ne 0$.

Definition 1

The greatest common divisor of $a$ and $b$ is defined as:

the largest $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$

Definition 2

The greatest common divisor of $a$ and $b$ is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that:

$(1): \quad d \divides a \land d \divides b$

$(2): \quad c \divides a \land c \divides b \implies c \divides d$

This is denoted $\gcd \set {a, b}$.

When $a = b = 0$, $\gcd \set {a, b}$ is undefined.