Multiples

A multiple is an arithmetic term that primarily refers to the multiplication of whole numbers ( ). However, it can also be generalized to any Abelian group . In the fractions and number theory that plays least common multiple of two or more integers a role. ${\displaystyle \dotsc ,-1,0,1,2,\dotsc }$

Multiple of an integer

An integer is called a multiple of a number if there is an integer such that is. In this case is a factor of . A real multiple is used when is. ${\displaystyle a\in \mathbb {Z} }$ ${\displaystyle b\in \mathbb {Z} }$ ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle a=k\cdot b}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle |k|>1}$

The multiple is also used for rational or real numbers and . ${\displaystyle a}$ ${\displaystyle b}$

Examples

• 6, 9, and −15 are proper multiples of 3.
• 6 is also a real multiple of 2.
• 0 is a multiple of any number.

Multiples in Abelian groups

If there is an Abelian group , there is a uniquely determined group homomorphism for each . For this applies ${\displaystyle (A,+)}$ ${\displaystyle a\in A}$ ${\displaystyle \mu _{a}(1)=a}$ ${\displaystyle \mu _{a}\colon \mathbb {Z} \to A}$

${\displaystyle {\begin{aligned}\mu _{a}(0)&=0,\\\mu _{a}(2)&=\mu _{a}(1)+\mu _{a}(1)=a+a,\\\mu _{a}(-n)&=-\mu _{a}(n)\ {\mbox{für alle}}\ n\in \mathbb {N} \,.\end{aligned}}}$

Well will

${\displaystyle n\cdot a:=\mu _{a}(n)}$

for and one calls a multiple or more precisely the -fold of in the Abelian group . ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle a\in A}$ ${\displaystyle n\cdot a}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle A}$

Web links

Wiktionary: Multiples  - explanations of meanings, word origins, synonyms, translations

• Multiple . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ Eric W. Weisstein : Multiple . In: MathWorld (English).
2. ↑ multiples . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .