# Definition:Greatest Common Divisor/Integers/General Definition

## Definition

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).

Then 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.

## 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

- Greatest Common Divisor is Associative for a justification of this construction.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $10$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 23 \theta$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 12$: Highest common factors and Euclid's algorithm