Equivalence

If (a mod n) = (b mod n), then a is equivalent to b, modulo n, denoted a $\equiv$ b (mod n).

An equivalence class modulo n containing an integer a
is [a]_n = {a + kn $\mid$ k $\in$ Z}.

Example: if a=8, n=3, then q = 2, r = 2, and some b $\equiv$ a are 2, 5, 8, 11, 14, 17, ....
The equivalence classes modulo n are

[0]₃ = {..., -3, 0, 3, 6, 9, 12, ...} =

[1]₃ = {..., -2, 1, 4, 7, 10, 13, ...}

[2]₃ = {..., -1, 2, 5, 8, 11, 14, ...}