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{{short description|Computation modulo a fixed integer}}
{{About|the concept that uses the "''{{mvar|a}} (mod {{mvar|m}})''" notation|the binary operation ''mod({{mvar|a,m}})'' |
{{More citations needed|date=June 2025}}
[[File:Clock group.svg|thumb|upright=1.1|right|Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12.]]
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\end{align}</math>
== Basic properties ==
{{anchor|Properties}}
The congruence relation satisfies all the conditions of an [[equivalence relation]]:
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=== 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 4.
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