Congruences and Modular Arithmetic

Congruences and Modular ArithmeticRyan C. DailedaTrinity UniversityNumber TheoryDailedaCongruencesIntroductionModu lar Arithmetic is the Arithmetic of remainders. The somewhat surprising fact is that Modular Arithmetic obeysmost of the same laws that ordinary Arithmetic explains, for instance, homework exercise on theassociativity of will later see that because of this the set of equivalence classesunder congruence moduloncan be given the structure of Nanda,b Z. We say thata is congruent to b modulon, denoteda b(modn), providedn|a have: 7 22 (mod 5), 4 3 (mod 7), 19 119(mod 100), 37 1 (mod 4).For anya,b Z:a b(mod 1).Notice that:a b(modn) a b=nk a=b+nkfor somek first result concerning Congruences should be familiar fromIntro to 1 Let n N. Then congruence modulo n is an equivalence (Sketch).Leta,b,c :Sincen|0,a a(modn).Symmetry:Ifn|a b, thenn| (a b) =b a.

Modular arithmetic is the “arithmetic of remainders.” The somewhat surprising fact is that modular arithmetic obeys most of the same laws that ordinary arithmetic does. This explains, for instance, homework exercise 1.1.4 on the associativity of remainders. We will later see that because of this the set of equivalence classes

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  Modular, Arithmetic, Modular arithmetic

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