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, 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 ${\frac {b}{a}}$ 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:

$14=7\times 2$
$49=7\times 7$
$-21=7\times (-3)$
$0=7\times 0$
$3=7\times (3/7),\quad 3/7$ is not an integer
$-6=7\times (-6/7),\quad -6/7$ is not an integer.

Properties

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