Reduced residue system

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.^[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:

• {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, ... , r_φ(n)} is a reduced residue system modulo n with n > 2, then ${\displaystyle \sum r_{i}\equiv 0\!\!\!\!\mod n}$.
• If {r₁, r₂, ... , r_φ(n)} is a reduced residue system modulo n, and a is an integer such that gcd(a, n) = 1, then {ar₁, ar₂, ... , ar_φ(n)} is also a reduced residue system modulo n.^[3][4]

See also

• Complete residue system modulo m
• Multiplicative group of integers modulo n
• Congruence relation
• Euler's totient function
• Greatest common divisor
• Least residue system modulo m
• Modular arithmetic
• Number theory
• Residue number system

Notes

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

References

• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 71081766

External links

• Residue systems at PlanetMath
• Reduced residue system at MathWorld