In number theory, a congruence is an equation of the form "a ≡ b (mod m)", where "a" and "b" are integers and "m" is a positive integer (called the modulus). The congruence states that the difference "a - b" is an integer multiple of "m". The properties of congruences include:
Reflexivity: a ≡ a (mod m) for all integers "a".
Symmetry: if a ≡ b (mod m), then b ≡ a (mod m).
Transitivity: if a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m).
Linearity: (a + b) ≡ (c + d) (mod m) if a ≡ c (mod m) and b ≡ d (mod m).
Compatibility: if a ≡ b (mod m) and c ≡ d (mod m), then (a + c) ≡ (b + d) (mod m).
These properties make congruences into an equivalence relation, which means they can be used to partition the set of integers into equivalence classes (known as residue classes).
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