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{{Short description|Set of residue classes modulo n, relatively prime to n}}
Any subset ''R'' of the integers is called a '''reduced residue system''' modulo ''n'' if: In ], a ] ''R'' of the ] is called a '''reduced residue system modulo''' ''n'' if:


#gcd(''r'', ''n'') = 1 for each ''r'' contained in ''R''; #gcd(''r'', ''n'') = 1 for each ''r'' in ''R'',
#''R'' contains φ(''n'') elements; and #''R'' contains φ(''n'') elements,
#no two elements of ''R'' are congruent modulo ''n''.<ref>{{harvtxt|Long|1972|p=85}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=104}}</ref> #no two elements of ''R'' are ] modulo ''n''.<ref>{{harvtxt|Long|1972|p=85}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=104}}</ref>


Here <math>\varphi</math> denotes ]. Here φ denotes ].


A reduced residue system modulo ''n'' can be formed from a ] modulo ''n'' by removing all integers not relatively prime to ''n''. For example, a complete residue system modulo 12 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is {1,5,7,11}. The ] of this set can be calculated with the totient function: <math>\varphi(12) = 4</math>. Some other reduced residue systems modulo 12 are: A reduced residue system modulo ''n'' can be formed from a ] modulo ''n'' by removing all integers not ] to ''n''. For example, a complete residue system modulo 12 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The so-called ]s 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is {1, 5, 7, 11}. The ] of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are:


*{13,17,19,23} *{13,17,19,23}
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==Facts== ==Facts==
*Every number in a reduced residue system modulo ''n'' is a ] for the additive ] of integers modulo ''n''.
⚫ *If {{math|{''r''<sub>1</sub>, ''r''<sub>2</sub>, ... , ''r''<sub>φ(''n'')</sub>} }} is a reduced residue system with ''n'' > 2, then <math>\sum r_i \equiv 0 \pmod n</math>.
*Every number in a reduced residue system mod ''n'' is a generator for the additive group of integers modulo n. *A reduced residue system modulo ''n'' is a ] under multiplication modulo ''n''.
⚫ *If {''r''<sub>1</sub>, ''r''<sub>2</sub>, ... , ''r''<sub>φ(''n'')</sub>} is a reduced residue system modulo ''n'' with ''n'' > 2, then <math>\sum r_i \equiv 0\!\!\!\!\mod n</math>.
*If {''r''<sub>1</sub>, ''r''<sub>2</sub>, ... , ''r''<sub>φ(''n'')</sub>} is a reduced residue system modulo ''n'', and ''a'' is an integer such that gcd(''a'', ''n'') = 1, then {''ar''<sub>1</sub>, ''ar''<sub>2</sub>, ... , ''ar''<sub>φ(''n'')</sub>} is also a reduced residue system modulo ''n''.<ref>{{harvtxt|Long|1972|p=86}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=108}}</ref>


==See also== ==See also==
*] *]
*]
*] *]
*] *]
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==External links== ==External links==
* at PlanetMath * at PlanetMath
* at MathWorld * at MathWorld


] ]
] ]

{{numtheory-stub}}

Latest revision as of 19:42, 29 April 2024

Set of residue classes modulo n, relatively prime to n

In mathematics, a subset R of the integers is called a reduced residue system modulo n if:

  1. gcd(r, n) = 1 for each r in R,
  2. R contains φ(n) elements,
  3. no two elements of R are congruent modulo n.

Here φ denotes Euler's totient function.

A reduced residue system modulo n can be formed from a complete residue system modulo n by removing all integers not relatively prime to n. For example, a complete residue system modulo 12 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The so-called totatives 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is {1, 5, 7, 11}. The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are:

  • {13,17,19,23}
  • {−11,−7,−5,−1}
  • {−7,−13,13,31}
  • {35,43,53,61}

Facts

  • Every number in a reduced residue system modulo n is a generator for the additive group of integers modulo n.
  • A reduced residue system modulo n is a group under multiplication modulo n.
  • If {r1, r2, ... , rφ(n)} is a reduced residue system modulo n with n > 2, then ∑ r i ≡ 0 mod n {\displaystyle \sum r_{i}\equiv 0\!\!\!\!\mod n} .
  • If {r1, r2, ... , rφ(n)} is a reduced residue system modulo n, and a is an integer such that gcd(a, n) = 1, then {ar1, ar2, ... , arφ(n)} is also a reduced residue system modulo n.

See also

Notes

  1. Long (1972, p. 85)
  2. Pettofrezzo & Byrkit (1970, p. 104)
  3. Long (1972, p. 86)
  4. Pettofrezzo & Byrkit (1970, p. 108)

References

External links

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