# 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

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