Modular arithmetic: Difference between revisions

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The congruence relation is an [[equivalence relation]]. The [[equivalence class]] modulo {{mvar|m}} of an integer {{math|''a''}} is the set of all integers of the form {{math|''a'' + ''k m''}}, where {{mvar|k}} is any integer. It is called the '''congruence class''' or '''residue class''' of {{math|''a''}} modulo {{math|''m''}}, and may be denoted {{math|(''a'' mod ''m'')}}, or as {{math|{{overline|''a''}}}} or {{math|[''a'']}} when the modulus {{math|''m''}} is known from the context.

Each residue class modulo&nbsp;{{math|''m''}} contains exactly one integer in the range <math>0, ..., |{{math|''m''}}| - 1</math>. Thus, these <|{{math>|''m''}}|</math> integers are [[representative (mathematics)|representatives]] of their respective residue classes.

It is generally easier to work with integers than sets of integers; that is, the representatives most often considered, rather than their residue classes.