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Math135Lecture04StudentNotes

Math135Lecture04StudentNotes - Math 135 Lecture 4...

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Math 135: Lecture 4: Existential Quantifiers 1 GCDs, Certificates and the Extended Euclidean Algorithm Example 4.1. Compute gcd(1386 , 322). Use GCD WR : If a and b are integers not both zero, and q and r are integers such that a = qb + r , then gcd( a, b ) = gcd( b, r ). Since 1386 = 4 × 322 + 98 , gcd(1386 , 322) = gcd( ) . Since 322 = 3 × 98 + 28 , gcd( ) = gcd( ) . Since 98 = 3 × 28 + 14 , gcd( ) = gcd( ) . Since 28 = 2 × 14 + 0 , gcd( ) = gcd( ) . Since , the chain of equalities gives . Exercise 4.2. With the person beside you, randomly pick two positive integers and compute their gcd using the Euclidean Algorithm. Certificates A certificate of correctness consists of two parts. 1. a theorem which you have already proved and which relates to the problem in general 2. data which relates to this specific problem Theorem 4.3 (GCD Characterization Theorem ( GCD CT )) . If d is a positive common divisor of the integers a and b , and there exist integers x and y so that ax + by = d , then d = gcd( a, b ). Example 4.4. Claim: gcd(1386 , 322) = 14. Our certificate of correctness consists of the GCD Characterization Theorem and the integers d = 14, x = 10 and y = - 43. Note that 14 | 1386 and 14 | 322 and 1386 × 10+322 × ( - 43) = 14, so we can conclude that 14 = gcd(1386 , 322).
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