# Multiple (mathematics)

Short description: Product with an integer

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 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 not an integer
• $\displaystyle{ -6 = 7 \times (-6/7), \quad -6/7 }$ is not an integer.

## Properties

• 0 is a multiple of every number ($\displaystyle{ 0=0\cdot b }$).
• 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. WordNet lexicon database, Princeton University
2. WordReference.com
3. The Free Dictionary by Farlex
4. Dictionary.com Unabridged
5. Cambridge Dictionary Online
6. "Submultiple". Merriam-Webster. 2017.
7. "Submultiple". Oxford University Press. 2017.
8.   Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes