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Multiple (mathematics)

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Product with an integer

Look upmultiple orsubmultiple in Wiktionary, the free dictionary.

Inmathematics, amultiple is theproduct of any quantity and aninteger.^[1] In other words, for the quantitiesa andb, it can be said thatb is a multiple ofa ifb =na for some integern, which is called themultiplier. Ifa is notzero, this is equivalent to saying that $b/a$ is an integer.

Whena andb are both integers, andb is a multiple ofa, thena is called adivisor ofb. One says also thata dividesb. Ifa andb are not integers, mathematicians prefer generally to useinteger multiple instead ofmultiple, for clarification. In fact,multiple is used for other kinds of product; for example, apolynomialp is a multiple of another polynomialq if there exists third polynomialr such thatp =qr.

Examples

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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 suchintegers for 3 and −6. Each of the products listed below, and in particular, the products for 3 and −6, is theonly 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

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

Submultiple

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In some texts^[which?], "a is asubmultiple ofb" has the meaning of "a being aunit fraction ofb" (a=b/n) or, equivalently, "b being aninteger multiplen ofa" (b=n a). This terminology is also used withunits of measurement (for example by theBIPM^[2] andNIST^[3]), where aunit submultiple is obtained byprefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 10³. For example, amillimetre is the 1000-fold submultiple of ametre.^[2]^[3] As another example, oneinch may be considered as a 12-fold submultiple of afoot, or a 36-fold submultiple of ayard.