# Definition:Common Divisor

## Definition

### Integral Domain

Let $\struct {D, +, \times}$ be an integral domain.

Let $S \subseteq D$ be a finite subset of $D$.

Let $c \in D$ such that $c$ divides all the elements of $S$, that is:

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

Then $c$ is a common divisor (or common factor) of all the elements in $S$.

### Integers

The definition is usually applied when the integral domain in question is the set of integers $\Z$, thus:

Let $S$ be a finite set of integers, that is:

$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \Z}$

Let $c \in \Z$ such that $c$ divides all the elements of $S$, that is:

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

Then $c$ is a common divisor (or common factor) of all the elements in $S$.

### Real Numbers

The definition can also be applied when the integral domain in question is the real numbers $\R$, thus:

Let $S$ be a finite set of real numbers, that is:

$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \R}$

Let $c \in \R$ such that $c$ divides all the elements of $S$, that is:

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

Then $c$ is a common divisor of all the elements in $S$.