Reduced residue system

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

Contents

  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. [1] [2]

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:

Facts

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)

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