Modular arithmetic: Difference between revisions

Revision as of 21:24, 19 December 2024

Revision as of 23:17, 30 December 2024

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where {{math|0 ≤ ''r'' < ''m''}} is the common remainder. We recover the previous relation ({{math|1=''a'' − ''b'' = ''k m''}}) by subtracting these two expressions and setting {{math|1=''k'' = ''p'' − ''q''.}}

Because the congruence modulo {{mvar|m}} is defined by the [[Divisor#Further notions and facts|divisibility]] by {{mvar|m}} and because {{math|-1−1}} is a [[Unit (ring theory)#Integer ring|unit]] in the ring of integers, a number is divisible by {{math|-''m''}} exactly if it is divisible by {{mvar|m}}.

This means that every non-zero integer {{mvar|m}} may be taken as modulus.

=== Examples ===