# Definition:Multiple

## Definition

Let $\struct {D, +, \circ}$ be an integral domain and $x, y \in D$.

If $x \divides y$, 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.

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

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 6$: The division process in $I$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary