# Multiple (mathematics)

__: Product with an integer__

**Short description**In science, a **multiple** is the product of any quantity and an integer.^{[1]}^{[2]}^{[3]} 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 *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.

## Examples

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:

- [math]\displaystyle{ 14 = 7 \times 2 }[/math]
- [math]\displaystyle{ 49 = 7 \times 7 }[/math]
- [math]\displaystyle{ -21 = 7 \times (-3) }[/math]
- [math]\displaystyle{ 0 = 7 \times 0 }[/math]
- [math]\displaystyle{ 3 = 7 \times (3/7), \quad 3/7 }[/math] is not an integer
- [math]\displaystyle{ -6 = 7 \times (-6/7), \quad -6/7 }[/math] is not an integer.

## Properties

- 0 is a multiple of every number ([math]\displaystyle{ 0=0\cdot b }[/math]).
- The product of any integer [math]\displaystyle{ n }[/math] and any integer is a multiple of [math]\displaystyle{ n }[/math]. In particular, [math]\displaystyle{ n }[/math], which is equal to [math]\displaystyle{ n \times 1 }[/math], is a multiple of [math]\displaystyle{ n }[/math] (every integer is a multiple of itself), since 1 is an integer.
- If [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are multiples of [math]\displaystyle{ x }[/math] then [math]\displaystyle{ a+b }[/math] and [math]\displaystyle{ a-b }[/math] are also multiples of [math]\displaystyle{ x }[/math].

## References

- ↑ Weisstein, Eric W.. "Multiple". http://mathworld.wolfram.com/Multiple.html.
- ↑ WordNet lexicon database, Princeton University
- ↑ WordReference.com
- ↑ The Free Dictionary by Farlex
- ↑ Dictionary.com Unabridged
- ↑ Cambridge Dictionary Online
- ↑ "Submultiple". Merriam-Webster. 2017. https://www.merriam-webster.com/dictionary/submultiple.
- ↑ "Submultiple". Oxford University Press. 2017. https://en.oxforddictionaries.com/definition/us/submultiple.
- ↑
^{9.0}^{9.1}International Bureau of Weights and Measures (2006),*The International System of Units (SI)*(8th ed.), ISBN 92-822-2213-6, http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf - ↑
^{10.0}^{10.1}"NIST Guide to the SI". http://physics.nist.gov/Pubs/SP811/sec04.html. Section 4.3:*Decimal multiples and submultiples of SI units: SI prefixes*

## See also

Original source: https://en.wikipedia.org/wiki/ Multiple (mathematics).
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