This chapter introduces the basic properties of congruences modulo n, along with the related notion of residue classes modulo n. Other items discussed include the Chinese remainder theorem, Euler's phi function, Euler's theorem, Fermat's little theorem, quadratic residues, and finally, summations over divisors.

Equivalence relations

Before discussing congruences, we review the definition and basic properties of equivalence relations.

Let S be a set. A binary relation ∼ on S is called an equivalence relation if it is

reflexive:a ∼ a for all a ∈ S,

symmetric:a ∼ b implies b ∼ a for all a, b ∈ S, and

transitive:a ∼ b and b ∼ c implies a ∼ c for all a, b, c ∈ S.

If ∼ is an equivalence relation on S, then for a ∈ S one defines its equivalence class as the set {x ∈ S : x ∼ a}.

Theorem 2.1. Let ∼ be an equivalence relation on a set S, and for a ∈ S, let [a] denote its equivalence class. Then for all a, b ∈ S, we have:

1. (i) a ∈ [a];

2. (ii) a ∈ [b] implies [a] = [b].

Proof. (i) follows immediately from reflexivity. For (ii), suppose a ∈ [b], so that a ∼ b by definition. We want to show that [a] = [b]. To this end, consider any x ∈ S.