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Any subset R of the set of integers is called a reduced residue system modulo n if

1. (r, n) = 1 for each r contained in R;
2. R contains ${\displaystyle \varphi }$(n) elements;
3. no two elements of R are congruent modulo n.

Here ${\displaystyle \varphi }$ 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}. 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}. Note that the cardinality of this set is ${\displaystyle \varphi (12)=4}$. 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

• If ${\displaystyle \{r_{1},r_{2},\dots ,r_{\varphi (n)}\}}$ is a reduced residue system with n > 2, then ${\displaystyle \sum r_{i}\equiv 0{\pmod {n}}}$.
• Every number in a reduced residue system mod n (except for 1) is a generator for the multiplicative group of integers mod n.

See also

• Congruence relation
• Euler's totient function
• Modular arithmetic
• Number theory
• Residue number system

External links

• Residue systems at PlanetMath
• Reduced residue system at MathWorld

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