congruence class (another Prime Pages' Glossary entries) Glossary: Prime Pages: Top 5000: GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)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 Chris K. Caldwell © 1999-2020 (all rights reserved)