# Definition:Greatest Common Divisor/Integers/Definition 1

## Definition

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

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

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

where $\divides$ denotes divisibility.

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

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

### General Definition

This definition can be extended to any (finite) number of integers.

Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero).

### Definition 1

The greatest common divisor of $S$:

$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$

is defined as the largest $d \in \Z_{>0}$ such that:

$\forall x \in S: d \divides x$

where $\divides$ denotes divisibility.

### Definition 2

The greatest common divisor of $S$:

$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$

is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that:

 $\ds \forall x \in S:$ $\ds d \divides x$ $\ds \forall e \in \Z: \forall x \in S:$ $\ds e \divides x \implies e \divides d$

where $\divides$ denotes divisibility.

By convention:

$\map \gcd \O = 1$

## Also defined as

Some sources gloss over the fact that at least one of $a$ and $b$ must be non-zero for $\gcd \set{ a, b }$ to be defined.

Some sources insist that both $a$ and $b$ be non-zero or strictly positive.

Some sources define $\gcd \set {a, b} = 0$ for $a = b = 0$.

## Also known as

The greatest common divisor is often seen abbreviated as GCD, gcd or g.c.d.

Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs.

The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable.

The greatest common divisor is also known as the highest common factor, or greatest common factor.

Highest common factor when it occurs, is usually abbreviated as HCF, hcf or h.c.f.

It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$.

The archaic term greatest common measure can also be found, mainly in such as Euclid's The Elements.

## Also see

• Results about the greatest common divisor can be found here.