congruence class

In our entry on congruences we note that if m is not zero and a, b and c are any integers, then we have the following:

The reflexive property: If a is any integer, a ≡ a (mod m),
The symmetric property: If a ≡ b (mod m), then b ≡ a (mod m),
The transitive property: If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m).

These three properties are just what we need to show that the integers are divided into exactly m congruence classes containing integers mutually congruent modulo m. (Technically, we say congruence is an equivalence relation.) For example, modulo five we have the 5 classes

... ≡ -10 ≡ -5 ≡ 0 ≡ 5 ≡ 10 ≡ 15 ≡ ... (mod 5)
... ≡ -9 ≡ -4 ≡ 1 ≡ 6 ≡ 11 ≡ 16 ≡ ... (mod 5)
... ≡ -8 ≡ -3 ≡ 2 ≡ 7 ≡ 12 ≡ 17 ≡ ... (mod 5)
... ≡ -7 ≡ -2 ≡ 3 ≡ 8 ≡ 13 ≡ 18 ≡ ... (mod 5)
... ≡ -6 ≡ -1 ≡ 4 ≡ 9 ≡ 14 ≡ 19 ≡ ... (mod 5)

Modulo two there are the two classes we call the even and odd integers:

... ≡ -4 ≡ -2 ≡ 0 ≡ 2 ≡ 4 ≡ 6 ≡ ... (mod 2)
... ≡ -3 ≡ -1 ≡ 1 ≡ 3 ≡ 5 ≡ 7 ≡ ... (mod 2)

Sometimes we denote these classes as 0 mod 2, and 1 mod 2 respectively.

See Also: Residue