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Difference between revisions of "Complete system of residues"

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''modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239102.png" />''
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''modulo $m$''
  
Any set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239103.png" /> integers that are incongruent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239104.png" />. Usually, as a complete residue system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239105.png" /> one takes the least non-negative residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239106.png" />, or the absolutely least residues consisting of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239107.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239108.png" /> is odd or the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c0239109.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023910/c02391010.png" /> is even.
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Any set of $m$ integers that are incongruent $\bmod\,m$. Usually, as a complete residue system $\bmod\,m$ one takes the least non-negative residues $0,\ldots,m-1$, or the absolutely least residues consisting of the number $0,\pm1,\ldots,\pm(m-1)/2$ if $m$ is odd or the numbers $0,\pm1,\ldots,\pm(m-2)/2,m/2$ if $m$ is even.
  
  

Revision as of 13:59, 11 August 2014

modulo $m$

Any set of $m$ integers that are incongruent $\bmod\,m$. Usually, as a complete residue system $\bmod\,m$ one takes the least non-negative residues $0,\ldots,m-1$, or the absolutely least residues consisting of the number $0,\pm1,\ldots,\pm(m-1)/2$ if $m$ is odd or the numbers $0,\pm1,\ldots,\pm(m-2)/2,m/2$ if $m$ is even.


Comments

See also Reduced system of residues.

How to Cite This Entry:
Complete system of residues. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_system_of_residues&oldid=32835
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article