Reduced residue system
# Reduced residue system

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

* gcd(r, n) = 1 for each r in R,
* R contains φ(n) elements,
* 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

* If ${r_1,$ $r_2,$ ... , $r_{φ(n)}}$ is a reduced residue system modulo n with n > 2, then $\sum r_{i}\equiv 0\!\!\!\!\mod n$.
* Every number in a reduced residue system modulo n is a generator for the additive group of integers modulo n.
* If ${r_1,$ $r_2,$ ... , $r_{φ(n)}}$ is a reduced residue system modulo n, and a is an integer such that gcd(a, n) = 1, then ${ar_1,$ $ar_2,$ ... , $ar_{φ(n)}}$ is also a reduced residue system modulo n.
