congruence class
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: 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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