# Multiple (mathematics)

*or*

**multiple***in Wiktionary, the free dictionary.*

**submultiple**In mathematics, a **multiple** is the product of any quantity and an integer.^{[1]} In other words, for the quantities *a* and *b*, it can be said that *b* is a multiple of *a* if *b* = *na* for some integer *n*, which is called the multiplier. If *a* is not zero, this is equivalent to saying that is an integer.

When *a* and *b* are both integers, and *b* is a multiple of *a*, then *a* is called a divisor of *b*. One says also that *a* divides *b*. If *a* and *b* are not integers, mathematicians prefer generally to use **integer multiple** instead of *multiple*, for clarification. In fact, *multiple* is used for other kinds of product; for example, a polynomial *p* is a multiple of another polynomial *q* if there exists third polynomial *r* such that *p* = *qr*.

In some texts, "*a* is a **submultiple** of *b*" has the meaning of "*a* being a unit fraction of *b*" or, equivalently, "*b* being an integer multiple of *a*". This terminology is also used with units of measurement (for example by the BIPM^{[2]} and NIST^{[3]}), where a *submultiple* of a main unit is a unit, named by prefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 10^{3}. For example, a millimetre is the 1000-fold submultiple of a metre.^{[2]}^{[3]} As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.

## Examples[edit]

14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such *integers* for 3 and −6. Each of the products listed below, and in particular, the products for 3 and −6, is the *only* way that the relevant number can be written as a product of 7 and another real number:

- is not an integer;
- is not an integer.

## Properties[edit]

- 0 is a multiple of every number ().
- The product of any integer and any integer is a multiple of . In particular, , which is equal to , is a multiple of (every integer is a multiple of itself), since 1 is an integer.
- If and are multiples of then and are also multiples of .

## See also[edit]

## References[edit]

**^**Weisstein, Eric W. "Multiple".*MathWorld*.- ^
^{a}^{b}International Bureau of Weights and Measures (2006),*The International System of Units (SI)*(PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16. - ^
^{a}^{b}"NIST Guide to the SI". Section 4.3:*Decimal multiples and submultiples of SI units: SI prefixes*.