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week05 - Congruences(Part 3 Intro to Rational Numbers...

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Congruences (Part 3) & Intro to Rational Numbers Original Notes adopted fromOctober 9, 2001 (Week 5) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong a & b are relatively prime if their only common factor is 1 Lemma : If m | ab and m & a are relatively prime, then m | b Fix m > 1 (m) = number of numbers in { 1,2,..., m – 1} that are relatively prime with m. If m is prime, (m) = p –1. Eg. m = 6 {1,2,3,4,5} (6) = 2... (have no common factors with 6) Lemma: If a & and m are relatively prime & b and m are relatively prime, then ab & m are relatively prime. Opposite of Relatively Prime: Have a common factor that is prime. Proof: If not, there exists p prime such that p |m and p |ab p|a or p |b. If p |a, a & m not relatively prime. If p |b, b & m not relatively prime .... Contradiction. Note: If a b (mod m ) and a & m are relatively prime, then b & m are relatively prime. For a = b + tm; some t Z. Euler's Theorem : For any m > 1, if a is relatively prime to m, then a m 1 (mod m).
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