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

• Definition:Divisor (Algebra)
• Definition:Common Multiple

Euclid's Definition

In the words of Euclid:

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

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

... and again:

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

(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.

(The Elements: Book $\text{VII}$: Definition $17$)

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiple
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiple