L05_InversesGCDs

L05_InversesGCDs - 1-1COMP170Discrete Mathematical Toolsfor...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 2.2, pp. 56-69Inverses and GCDsVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-12.2 Inverses and Greatest Common Divisors•Solutions to Equations and Inverses modn•Converting Modular Equations to Normal Equations•Greatest Common Divisors•Euclid’s Division Theorem•Euclid’s GCD Algorithm•Extended GCD Agorithm•Computing Inverses3-1Greatest CommonDivisors:DefinitionsEuclid’sDivisionTheoremEuclid’sGCDAlgorithmSolutions toEqns & InversesmodnConvertingmodnEqns to NormalOnesExtendedGCDAlgorithmComputingInversesmodnProving toolfrom prev lecture3-2Greatest CommonDivisors:DefinitionsEuclid’sDivisionTheoremEuclid’sGCDAlgorithmSolutions toEqns & InversesmodnConvertingmodnEqns to NormalOnesExtendedGCDAlgorithmComputingInversesmodnProving toolfrom prev lectureThis lecture develops lots of tools forlater use. Here’s a chart describing therelationship between the various parts3-3Greatest CommonDivisors:DefinitionsEuclid’sDivisionTheoremEuclid’sGCDAlgorithmSolutions toEqns & InversesmodnConvertingmodnEqns to NormalOnesExtendedGCDAlgorithmComputingInversesmodnProving toolfrom prev lectureThis lecture develops lots of tools forlater use. Here’s a chart describing therelationship between the various partsIntroduction toproof by smallestcounterexampleIntroduction toproof bycontradiction4-1Definition:•Positive integermis adivisorof integernifn=mqfor some integerq•ifmis a divisor ofnwe writem|n.(say) “mdividesn”•ifmis anota divisor ofnwe writem6 |n.(say) “mdoes not dividen”4-2Definition:•Positive integermis adivisorof integernifn=mqfor some integerq•ifmis a divisor ofnwe writem|n.(say) “mdividesn”•ifmis anota divisor ofnwe writem6 |n.(say) “mdoes not dividen”Examples:•1|30,5|30,5|35,56 |315-1•Ifpis a divisor ofbothmandnthenpis acommon divisorofmandn•gcd(m,n)denotes thegreatest common divisorofmandn.1is aways a common divisor ofmandnDefinition:5-2•Ifpis a divisor ofbothmandnthenpis acommon divisorofmandn•gcd(m,n)denotes thegreatest common divisorofmandn.1is aways a common divisor ofmandnDefinition:Examples:•{1,2,3,6}areallof the common divisors of24and30.•gcd(24,30) = 66-1Definition:•Positive integerp >1isprimeif its only divisors are1anditself .Ifpis not prime, it iscomposite.•mandnarerelatively primeif they have no commondivisor other than1, i.e.,gcd(m,n) = 1.6-2Definition:•Positive integerp >1isprimeif its only divisors are1anditself .Ifpis not prime, it iscomposite.•mandnarerelatively primeif they have no commondivisor other than1, i.e.,gcd(m,n) = 1....
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L05_InversesGCDs - 1-1COMP170Discrete Mathematical Toolsfor...

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