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