Editing Modular arithmetic (section)

== Residue systems ==
Each residue class modulo {{math|''m''}} may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Modular Arithmetic|url=https://mathworld.wolfram.com/ModularArithmetic.html|access-date=2020-08-12|website=Wolfram MathWorld|language=en|archive-date=2023-07-14|archive-url=https://web.archive.org/web/20230714132828/https://mathworld.wolfram.com/ModularArithmetic.html|url-status=live}}</ref> (since this is the proper remainder which results from division). Any two members of different residue classes modulo {{math|''m''}} are incongruent modulo {{math|''m''}}. Furthermore, every integer belongs to one and only one residue class modulo {{math|''m''}}.<ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=90}}</ref>

The set of integers {{math|{{mset|0, 1, 2, ..., ''m'' − 1}}}} is called the '''least residue system modulo {{math|''m''}}'''. Any set of {{math|''m''}} integers, no two of which are congruent modulo {{math|''m''}}, is called a '''complete residue system modulo {{math|''m''}}'''.

The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one [[representative (mathematics)|representative]] of each residue class modulo {{math|''m''}}.<ref>{{harvtxt|Long|1972|p=78}}</ref> For example, the least residue system modulo {{math|4}} is {{math|{{mset|0, 1, 2, 3}}}}. Some other complete residue systems modulo {{math|4}} include:
* {{math|{{mset|1, 2, 3, 4}}}}
* {{math|{{mset|13, 14, 15, 16}}}}
* {{math|{{mset|−2, −1, 0, 1}}}}
* {{math|{{mset|−13, 4, 17, 18}}}}
* {{math|{{mset|−5, 0, 6, 21}}}}
* {{math|{{mset|27, 32, 37, 42}}}}

Some sets that are ''not'' complete residue systems modulo 4 are:
* {{math|{{mset|−5, 0, 6, 22}}}}, since {{math|6}} is congruent to {{math|22}} modulo {{math|4}}.
* {{math|{{mset|5, 15}}}}, since a complete residue system modulo {{math|4}} must have exactly {{math|4}} incongruent residue classes.<!-- This example is used in the following subsection so please do not alter it. -->

=== Reduced residue systems ===
{{main|Reduced residue system}}
Given the [[Euler's totient function]] {{math|''φ''(''m'')}}, any set of {{math|''φ''(''m'')}} integers that are [[Coprime integers|relatively prime]] to {{math|''m''}} and mutually incongruent under modulus {{math|''m''}} is called a '''reduced residue system modulo {{math|''m''}}'''.<ref>{{harvtxt|Long|1972|p=85}}</ref> The set {{math|{{mset|5, 15}}}} from above, for example, is an instance of a reduced residue system modulo&nbsp;4.

=== Covering systems ===
{{main|Covering system}}
Covering systems represent yet another type of residue system that may contain residues with varying moduli.