# Multiple (mathematics)

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, 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 103. 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:

• ${\displaystyle 14=7\times 2}$
• ${\displaystyle 49=7\times 7}$
• ${\displaystyle -21=7\times (-3)}$
• ${\displaystyle 0=7\times 0}$
• ${\displaystyle 3=7\times (3/7),\quad 3/7}$ is a rational number, not an integer
• ${\displaystyle -6=7\times (-6/7),\quad -6/7}$ is a rational number, not an integer.

## Properties

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

## References

1. ^ Weisstein, Eric W. "Multiple". MathWorld.
2. ^ WordNet lexicon database, Princeton University
3. ^ WordReference.com
4. ^ The Free Dictionary by Farlex
5. ^ Dictionary.com Unabridged
6. ^ Cambridge Dictionary Online
7. ^ "Submultiple". Merriam-Webster Online Dictionary. Merriam-Webster. 2017. Retrieved 2017-02-01.
8. ^ "Submultiple". Oxford Living Dictionaries. Oxford University Press. 2017. Retrieved 2017-02-01.
9. ^ 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
10. ^ a b "NIST Guide to the SI". Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes