# Multiple (mathematics)

Look up or multiple in Wiktionary, the free dictionary.submultiple |

In mathematics, a **multiple** is the product of any quantity and an integer.^{[1]}^{[2]}^{[3]} In other words, for the quantities *a* and *b*, we say 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 *b*/*a* is an integer.^{[4]}^{[5]}^{[6]}

In mathematics, 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 "*b* being an integer multiple of *a*".^{[7]}^{[8]} This terminology is also used with units of measurement (for example by the BIPM^{[9]} and NIST^{[10]}), 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.^{[9]}^{[10]} As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.

## ExamplesEdit

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.

## PropertiesEdit

- 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 .

## ReferencesEdit

**^**Weisstein, Eric W. "Multiple".*MathWorld*.**^**WordNet lexicon database, Princeton University**^**WordReference.com**^**The Free Dictionary by Farlex**^**Dictionary.com Unabridged**^**Cambridge Dictionary Online**^**"Submultiple".*Merriam-Webster Online Dictionary*. Merriam-Webster. 2017. Retrieved 2017-02-01.**^**"Submultiple".*Oxford Living Dictionaries*. Oxford University Press. 2017. Retrieved 2017-02-01.- ^
^{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 2017-08-14 - ^
^{a}^{b}"NIST Guide to the SI". Section 4.3:*Decimal multiples and submultiples of SI units: SI prefixes*