280wk5_x4

# 280wk5_x4 - Example of Extended Euclidean Algorithm Recall...

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Example of Extended Euclidean Algorithm Recall that gcd(84 , 33) = gcd(33 , 18) = gcd(18 , 15) = gcd(15 , 3) = gcd(3 , 0) = 3 We work backwards to write 3 as a linear combination of 84 and 33: 3 = 18 - 15 [Now 3 is a linear combination of 18 and 15] = 18 - (33 - 18) = 2(18) - 33 [Now 3 is a linear combination of 18 and 33] = 2(84 - 2 × 33)) - 33 = 2 × 84 - 5 × 33 [Now 3 is a linear combination of 84 and 33] 1 Some Consequences Corollary 2: If a and b are relatively prime, then there exist s and t such that as + bt = 1. Corollary 3: If gcd( a,b ) = 1 and a | bc , then a | c . Proof: Exist s, t Z such that sa + tb = 1 Multiply both sides by c : sac + tbc = c Since a | bc , a | sac + tbc , so a | c Corollary 4: If p is prime and p | Π n i =1 a i , then p | a i for some 1 i n . Proof: By induction on n : If n = 1: trivial. Suppose the result holds for n and p | Π n +1 i =1 a i . note that p | Π n +1 i =1 a i = (Π n i =1 a i ) a n +1 . If p | a n +1 we are done. If not, gcd( p,a n +1 ) = 1. By Corollary 3, p | Π n i =1 a i By the IH, p | a i for some 1 i n . 2 The Fundamental Theorem of Arithmetic, II Theorem 3: Every n > 1 can be represented uniquely as a product of primes, written in nondecreasing size. Proof: Still need to prove uniqueness. We do it by strong induction. Base case: Obvious if n = 2. Inductive step. Suppose OK for n p < n . Suppose that n = Π s i =1 p i = Π r j =1 q j . p 1 | Π r j =1 q j , so by Corollary 4, p 1 | q j for some j . But then p 1 = q j , since both p 1 and q j are prime. But then n/p 1 = p 2 ··· p s = q 1 ··· q j - 1 q j +1 ··· q r Result now follows from I.H. 3 Characterizing the GCD and LCM Theorem 6: Suppose a = Π n i =1 p α i i and b = Π n i =1 p β i i , where p i are primes and α i , β i N . Some α i ’s, β i ’s could be 0. Then gcd( a,b ) = Π n i =1 p min( α i i ) i lcm( a,b ) = Π n i =1 p max( α i i ) i Proof: For gcd, let c = Π n i =1 p min( α i i ) i . Clearly c | a and c | b . Thus, c is a common divisor, so c gcd( a,b ). If q γ | gcd( a,b ), must have q ∈ { p 1 , .. ., p n } Otherwise q n | a so q n | gcd( a,b ) (likewise b ) If q = p i , q γ | gcd( a,b ), must have γ min( α i , β i ) E.g., if γ > α i , then p γ i n | a Thus, c gcd( a,b ). Conclusion: c = gcd( a,b ). 4

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For lcm, let d = Π n i =1 p max( α i i ) i . Clearly a | d , b | d , so d is a common multiple. Thus, d lcm( a,b ). Suppose lcm( a,b ) = Π n i =1 p γ i i . Must have α i γ i , since p α i i | a and a | lcm( a,b ). Similarly, must have β i γ i . Thus, max( α i i ) γ i .
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280wk5_x4 - Example of Extended Euclidean Algorithm Recall...

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