# Definition:Multiple

## Definition

### Integral Domain

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

Let $x, y \in D$.

Let $x$ be a divisor of $y$.

Then $y$ is a **multiple** of $x$.

### Integers

As the set of integers form an integral domain, the concept of being a **multiple** is fully applicable to the integers.

Let $\Z$ denote the set of integers.

Let $x, y \in \Z$.

Let $x$ be a divisor of $y$.

Then $y$ is a **multiple** of $x$.

## Also see

## Euclid's Definition

In the words of Euclid:

(*The Elements*: Book $\text{V}$: Definition $2$)

... and again:

(*The Elements*: Book $\text{VII}$: Definition $5$)

*A number is said to***multiply**a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

(*The Elements*: Book $\text{VII}$: Definition $15$)

*And, when two numbers having multiplied one another make some number, the number so produced is called***plane**, and its**sides**are the numbers which have multiplied one another.

(*The Elements*: Book $\text{VII}$: Definition $16$)

*And, when three numbers having multiplied one another make some number, the number so produced is***solid**, and its**sides**are the numbers which have multiplied one another.