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week04 - Congruences(Part 2 Fundamental Theorem of...

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Congruences (Part 2) & Fundamental Theorem of Arithmetic Original Notes adopted fromOctober 2, 2001 (Week 4) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Fundamental Theorem of Arithmetic Every natural number other than 1 can be represented as a product of primes and that representation is unique up to the order of factors. Eg. = 5 * 30 = 5 * 3 * 10 = 5 * 3* 5* 2 = 2 * 3* 5* 5 Given n, write n = p 1 α 1 , p 2 α 2 … p α k where each p i is prime and each α i is a natural number. Eg. 7 249 3 568 Eg. 12|36 (12 is not prime) 36 = 4 * 9, 12 doesn’t divide into 4, 12 doesn’t divide into 9. Theorem : If p is prime & a, b N, and if p | (ab) then p|a or p|b Proof: If a or b = 1, done. If not a = p 1 α 1 ….p s α s b = q 1 β 1 … q t β t ab = p 1 α 1 …. p s α s q 1 β 1 ….q t β t Given ab = pk, some k p occurs in factorization of ab into primes p = p i some i, or p = q j , some j If p = p i , then p|a If p = q j , then p |a Recall for m > 1 a A ± b (mod m) if m | a-b Eg. m = 5 (remainders of 0,1,2,34) Any a is congruent to 1 of {0,1,2,3,4} mod 5.
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