Content deleted Content added
use generalized version of a property. Tag: Reverted |
|||
(4 intermediate revisions by 4 users not shown) | |||
Line 1:
{{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.]]
Line 63 ⟶ 64:
* If {{math|''a'' + ''k'' ≡ ''b'' + ''k'' (mod ''m'')}}, where {{math|''k''}} is any integer, then {{math|''a'' ≡ ''b'' (mod ''m'')}}.
* If {{math|''k a'' ≡ ''k b'' (mod ''m'')}} and {{math|''k''}} is coprime with {{math|''m''}}, then {{math|''a'' ≡ ''b'' (mod ''m'')}}.
* If {{math|''k a'' ≡ ''k b'' (mod ''k m'')}} and {{math|''k'' ≠ 0}}, then {{math|''a'' ≡ ''b'' (mod ''m'')}}.
The last rule can be used to move modular arithmetic into division. If {{math|''b''}} divides {{math|''a''}}, then {{math|1=(''a''/''b'') mod ''m'' = (''a'' mod ''b m'') / ''b''}}.
Line 119 ⟶ 120:
=== 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.
|