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[parent] residue systems (Definition)

Definition. Let $ m$ be a positive integer. A complete residue system modulo $ m$ is a set $ \{a_0,\,a_1,\,\ldots,\,a_{m-1}\}$ of integers containing one and only one representant from every residue class modulo $ m$. Thus the numbers $ a_\nu$ may be ordered such that for all $ \nu = 0,\,1,\,\ldots,\,m\!-\!1$, they satisfy $ a_\nu \equiv \nu \pmod{m}$.

The least non-negative remainders modulo $ m$, i.e. the numbers

$\displaystyle 0,\,1,\,\ldots,\,m\!-\!1$
form a complete residue system modulo $ m$ (see long division). If $ m$ is odd, then also the absolutely least remainders
$\displaystyle -\frac{m\!-\!1}{2},\,-\frac{m\!-\!3}{2},\,\ldots,\, -2,\,-1,\,0,\,1,\,2,\,\ldots,\,\frac{m\!-\!3}{2},\,\frac{m\!-\!1}{2}$
offer a complete residue system modulo $ m$.

Theorem 1. If $ \gcd(a,\,m) = 1$ and $ b$ is an arbitrary integer, then the numbers

$\displaystyle 0a\!+\!b,\,1a\!+\!b,\,2a\!+\!b,\,\ldots,\,(m\!-\!1)a\!+\!b$
form a complete residue system modulo $ m$.

One may speak also of a reduced residue system modulo $ m$, which contains only one representant from each prime residue class modulo $ m$. Such a system consists of $ \varphi(m)$ integers, where $ \varphi$ means the Euler's totient function.

Remark. A set of integers forms a reduced residue system modulo $ m$ if and only if
$ 1^\circ$ it contains $ \varphi(m)$ numbers,
$ 2^\circ$ its numbers are coprime with $ m$,
$ 3^\circ$ its numbers are incongruent modulo $ m$.

Theorem 2. If $ \gcd(a,\,m) = 1$ and $ \{k_1,\,k_2,\,\ldots,\,k_{\varphi(m)}\}$ is a reduced residue system modulo $ m$, then also the numbers

$\displaystyle k_1a,\,k_2a,\,\ldots,\,k_{\varphi(a)}a$
form a reduced residue system modulo $ m$.

Bibliography

1
K. V¨AISÄLÄ: Lukuteorian ja korkeamman algebran alkeet. Otava, Helsinki (1950).



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See Also: prime residue class, totative, ${\mathbb{Z}}_n$

Also defines:  complete residue system, reduced residue system, least non-negative remainders, absolutely least remainders

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Cross-references: coprime, Euler's totient function, prime residue class, contains, odd, long division, numbers, residue class, integer, positive
There are 6 references to this entry.

This is version 8 of residue systems, born on 2007-04-19, modified 2008-01-30.
Object id is 9221, canonical name is ResidueSystems.
Accessed 2544 times total.

Classification:
AMS MSC: 11-00 (Number theory :: General reference works )
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