Editing Modular arithmetic (section)

== Congruence classes <span class="anchor" id="Residue"></span><span class="anchor" id="Residue class"></span><span class="anchor" id="Congruence class"></span> ==
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&nbsp;{{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, ..., |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. 

Consequently, {{math|(''a'' mod ''m'')}} denotes generally the unique integer {{mvar|r}} such that {{math|0 ≤ ''r'' < ''m''}} and {{math|''r'' ≡ ''a'' (mod ''m'')}}; it is called the '''residue''' of {{math|''a''}} modulo&nbsp;{{math|''m''}}.

In particular, {{math|1=(''a'' mod ''m'') = (''b'' mod ''m'')}} is equivalent to {{math|''a'' ≡ ''b'' (mod ''m'')}}, and this explains why "{{math|1==}}" is often used instead of "{{math|≡}}" in this context.