p| QED AlbertRMeyer March5,2012 lec5M. 11 AlbertRMeyer - PowerPoint PPT Presentation

Fundamental Thm. of Arithmetic Every integer > 1 Prime factors uniquely into a weakly decreasing Factorization sequence of primes Albert R Meyer March 5, 2012 lec 5M. 9 Albert R Meyer March 5, 2012 lec 5M. 10 Unique Prime Factorization Prime Divisibility Lemma: p prime and p | ab Example: implies p|a or p|b 61394323221 = pf: say not(p|a), so gcd(p,a) = 1 sa b + tp b = 1 b 53 · 37 · 37 · 37 · 11 · 11 · 7 · 3 · 3 · 3 � so, � � p| so p| p| QED Albert R Meyer March 5, 2012 lec 5M. 11 Albert R Meyer March 5, 2012 lec 5M. 12 3

Prime Divisibility Unique Prime Factorization Cor :If p is prime, and Every integer n > 1 has a unique factorization into p|a 1 ·a 2 · ··· ·a m primes: p 1 · ··· ·p k = n then p|a i for some i. p 1 ≥ p 2 ≥ ··· ≥ p k with pf : by induction on m. Albert R Meyer March 5, 2012 lec 5M. 13 Albert R Meyer March 5, 2012 lec 5M. 14 Unique Prime Factorization Unique Prime Factorization pf: suppose not. choose smallest n > 1: pf: suppose not. choose smallest n > 1: n = p 1 ·p 2 ···p k = q 1 ·q 2 ···q m n = p 1 ·p 2 ···p k = q 1 ·q 2 ···q m p 1 ≥ p 2 ≥ ··· ≥ p k p 1 ≥ p 2 ≥ ··· ≥ p k q 1 ≥ q 2 ≥ ··· ≥ q m q 1 ≥ q 2 ≥ ··· ≥ q m So can assume q 1 > p 1 ≥ p i If q 1 = p 1 , then p 2 ···p k = q 2 ···q m is smaller nonunique. Albert R Meyer March 5, 2012 lec 5M. 15 Albert R Meyer March 5, 2012 lec 5M. 16 4

Unique Prime Factorization pf: but q 1 |n = p 1 ·p 2 ···p k so q 1 |p i for some i by Cor, contradicting that q 1 > p i QED Albert R Meyer March 5, 2012 lec 5M. 17 5

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