Question

what is the residue of 31 modules 7?

Answer 1

gcd(r, n) = 1 for each r in R,

R contains φ(n) elements,

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}