# 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 $x$.

## Euclid's Definition

In the words of Euclid:

The greater is a multiple of the less when it is measured by the less.

... and again:

The greater number is a multiple of the less when it is measured by the less.
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.
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.
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.