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L05_InversesGCDs - COMP170 Discrete Mathematical Tools for...

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1-1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 2.2, pp. 56-69 Inverses and GCDs Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen
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2-1 2.2 Inverses and Greatest Common Divisors Solutions to Equations and Inverses mod n Converting Modular Equations to Normal Equations Greatest Common Divisors Euclid’s Division Theorem Euclid’s GCD Algorithm Extended GCD Agorithm Computing Inverses
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3-1 Greatest Common Divisors: Definitions Euclid’s Division Theorem Euclid’s GCD Algorithm Solutions to Eqns & Inverses mod n Converting mod n Eqns to Normal Ones Extended GCD Algorithm Computing Inverses mod n Proving tool from prev lecture
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3-2 Greatest Common Divisors: Definitions Euclid’s Division Theorem Euclid’s GCD Algorithm Solutions to Eqns & Inverses mod n Converting mod n Eqns to Normal Ones Extended GCD Algorithm Computing Inverses mod n Proving tool from prev lecture This lecture develops lots of tools for later use. Here’s a chart describing the relationship between the various parts
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3-3 Greatest Common Divisors: Definitions Euclid’s Division Theorem Euclid’s GCD Algorithm Solutions to Eqns & Inverses mod n Converting mod n Eqns to Normal Ones Extended GCD Algorithm Computing Inverses mod n Proving tool from prev lecture This lecture develops lots of tools for later use. Here’s a chart describing the relationship between the various parts Introduction to proof by smallest counterexample Introduction to proof by contradiction
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4-1 Definition: Positive integer m is a divisor of integer n if n = mq for some integer q if m is a divisor of n we write m | n . (say) “ m divides n if m is a not a divisor of n we write m | n . (say) “ m does not divide n
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4-2 Definition: Positive integer m is a divisor of integer n if n = mq for some integer q if m is a divisor of n we write m | n . (say) “ m divides n if m is a not a divisor of n we write m | n . (say) “ m does not divide n Examples: 1 | 30 , 5 | 30 , 5 | 35 , 5 | 31
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5-1 If p is a divisor of both m and n then p is a common divisor of m and n gcd ( m, n ) denotes the greatest common divisor of m and n . 1 is aways a common divisor of m and n Definition:
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5-2 If p is a divisor of both m and n then p is a common divisor of m and n gcd ( m, n ) denotes the greatest common divisor of m and n . 1 is aways a common divisor of m and n Definition: Examples: { 1 , 2 , 3 , 6 } are all of the common divisors of 24 and 30 . gcd (24 , 30) = 6
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6-1 Definition: Positive integer p > 1 is prime if its only divisors are 1 and itself . If p is not prime, it is composite . m and n are relatively prime if they have no common divisor other than 1 , i.e., gcd ( m, n ) = 1 .
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6-2 Definition: Positive integer p > 1 is prime if its only divisors are 1 and itself . If p is not prime, it is composite .
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