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CLASSROOM RESOURCE MATERIALS A n d a r l r i e d t n h o m W e t l i ac A n d y L i u Arithmetical Wonderland (cid:2)c 2015by TheMathematicalAssociationofAmerica,Inc. LibraryofCongressCatalogCardNumber2015950206 PrinteditionISBN978-0-88385-789-2 ElectroniceditionISBN978-1-61444-119-9 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 10 9 8 7 6 5 4 3 2 1 Arithmetical Wonderland Andrew C. F. Liu University of Alberta PublishedandDistributedby TheMathematicalAssociationofAmerica CouncilonPublicationsandCommunications JenniferJ.Quinn,Chair CommitteeonBooks FernandoGouveâ,Chair ClassroomResourceMaterialsEditorialBoard SusanG.Staples,Editor JenniferBergner CarenL.Diefenderfer ChristinaEubanks-Turner ChristopherHallstrom CynthiaJ.Huffman BrianPaulKatz PaulR.Klingsberg BrianLins MaryEugeniaMorley PhilipP.Mummert DarrylYong CLASSROOMRESOURCEMATERIALS Classroom Resource Materials is intended to provide supplementary classroom material for students—laboratory exercises, projects, historical information, textbooks with unusual ap- proachesforpresentingmathematicalideas,careerinformation,etc. 101CareersinMathematics,3rdeditioneditedbyAndrewSterrett Archimedes:WhatDidHeDoBesidesCryEureka?,ShermanStein ArithmeticalWonderland,AndrewC.F.Liu Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer,JohnMaceli,andEricRobinson CalculusMysteriesandThrillers,R.GrantWoods ConjectureandProof,Miklo´sLaczkovich CounterexamplesinCalculus,SergiyKlymchuk CreativeMathematics,H.S.Wall EnvironmentalMathematicsintheClassroom,editedbyB.A.FusaroandP.C.Kenschaft ExcursionsinClassicalAnalysis:PathwaystoAdvancedProblemSolvingandUndergraduate Research,byHongweiChen Explorations in Complex Analysis, Michael A. Brilleslyper, Michael J. Dorff, Jane M. McDougall, James S. Rolf, Lisbeth E. Schaubroeck, Richard L. Stankewitz, and Kenneth Stephenson ExploratoryExamplesforRealAnalysis,JoanneE.SnowandKirkE.Weller ExploringAdvancedEuclideanGeometrywithGeoGebra,GerardA.Venema GameTheoryThroughExamples,ErichPrisner GeometryFromAfrica:MathematicalandEducationalExplorations,PaulusGerdes TheHeartofCalculus:ExplorationsandApplications,PhilipAnseloneandJohnLee HistoricalModulesfortheTeachingandLearningofMathematics(CD),editedbyVictorKatz andKarenDeeMichalowicz IdentificationNumbersandCheckDigitSchemes,JosephKirtland InterdisciplinaryLivelyApplicationProjects,editedbyChrisArney InverseProblems:ActivitiesforUndergraduates,CharlesW.Groetsch Keeping it R.E.A.L.: Research Experiences for All Learners, Carla D. Martin and Anthony Tongen LaboratoryExperiencesinGroupTheory,EllenMaycockParker LearnfromtheMasters,FrankSwetz,JohnFauvel,OttoBekken,BengtJohansson,andVictor Katz MathMadeVisual:CreatingImagesforUnderstandingMathematics,ClaudiAlsinaandRoger B.Nelsen Mathematics Galore!: The First Five Years of the St. Marks Institute of Mathematics, James Tanton MethodsforEuclideanGeometry,OwenByer,FelixLazebnik,andDeirdreL.Smeltzer OrdinaryDifferentialEquations:ABriefEclecticTour,DavidA.Sańchez OvalTrackandOtherPermutationPuzzles,JohnO.Kiltinen ParadoxesandSophismsinCalculus,SergiyKlymchukandSusanStaples APrimerofAbstractMathematics,RobertB.Ash ProofsWithoutWords,RogerB.Nelsen ProofsWithoutWordsII,RogerB.Nelsen RediscoveringMathematics:YouDotheMath,ShaiSimonson SheDoesMath!,editedbyMarlaParker SolveThis:MathActivitiesforStudentsandClubs,JamesS.Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simulation, DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklic TeachingStatisticsUsingBaseball,JimAlbert VisualGroupTheory,NathanC.Carter WhichNumbersareReal?,MichaelHenle WritingProjectsforMathematicsCourses:CrushedClowns,Cars,andCoffeetoGo,Annalisa Crannell,GavinLaRose,ThomasRatliff,andElynRykken MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 1-800-331-1MAA FAX:1-301-206-9789 Contents PrefacetoaPreliminaryEdition ix Introduction xi 0 ReviewofArithmetic 1 0.1 CountingNumbers 1 0.2 Integers 13 0.3 Inequalities 18 0.4 Extras 23 1 Divisibility 27 1.1 BasicPropertiesofDivisibility 27 1.2 TheArithmeticofDivisibility 31 1.3 DivisibilityProblems 34 1.4 Extras 38 2 Congruence 45 2.1 TheDivisionAlgorithm 46 2.2 BasicPropertiesandArithmeticofCongruence 50 2.3 CongruenceandDivisibility 54 2.4 Extras 59 3 CommonDivisorsandMultiples 65 3.1 GreatestCommonDivisorsandtheEuclideanAlgorithm 66 3.2 RelativelyPrimeNumbers 73 3.3 LeastCommonMultiples 77 3.4 Extras 81 4 LinearDiophantineEquations 91 4.1 Be´zoutianAlgorithm 91 4.2 HomogeneousandNon-homogeneousEquations 99 4.3 LinearDiophantineProblems 102 4.4 Extras 108 vii viii Contents 5 PrimeFactorizations 115 5.1 PrimeandCompositeNumbers 116 5.2 FundamentalTheoremofArithmetic 119 5.3 ApplicationsoftheFundamentalTheoremofArithmetic 124 5.4 Extras 127 6 RationalandIrrationalNumbers 133 6.1 Fractions 133 6.2 Decimals 138 6.3 RealNumbers 144 6.4 Extras 149 7 NumerationSystems 155 7.1 ArithmeticinOtherBases 156 7.2 ConversionbetweenBases 163 7.3 ApplicationsofOtherBases 167 7.4 Extras 174 Appendix:ALegacyofMartinGardner 183 SolutiontoOdd-numberedExercises 185 Index 221 AbouttheAuthor 225 Preface to a Preliminary Edition by Martin Gardner Andy Liu, since 1980 a distinguished professor of mathematics at the University of Alberta, Edmonton,Canada,isoneoftheworld’sleadingexpertsinproblemsolving.Heisalsonoted for what has been called a “unique ability to present difficult concepts in a clear and logical manner”.Hisextremelybrightyoungpupils,ofbothgenders,areconstantlywinningprizesin toughmathematicscompetitions. Andy has won many awards, the most recent being the 2003 Adrien Pouliot Award from the Canadian Mathematical Society for his contribution to Canadian mathematics education, andthe2004DeborahandFranklinTepperHaimoAwardfromtheMathematicalAssociation ofAmericaforhisexcellenceinmathematicsteaching.HehaseditedaBookReviewcolumn forCruxMathematicorumandaProblemCornerforMathHorizons. Andyhasfinallydrawnuponhisvastexperienceasateacherofelementarymathematics toproducethetextbookyounowhold—abookthatcoverseverythingatalentedstudentwould wanttoknowaboutarithmetic.Thebook’swealthofstimulatingproblemsstartswithdivisibility tests—simple rules for every integer up to 12, except 7—to a final chapter on numeration systems.Alongthewayaretopicsnotusuallyfoundinsuchtextbooks,suchasthechapteron DiophantineEquations.Everyproblemisclearlystated,andeveryanswerandeveryproofeasy tounderstand. Andy is a great believer in using recreational mathematics to trigger a student’s interest. An admirer of mathematician/author Lewis Carroll, Andy sprinkled his book liberally with characters from Alice in Wonderland and Through the Looking Glass, especially in amusing dialoguesbetweenAliceandtheTweedletwins. ArithmeticalWonderlandisabookeveryteacheroftheelementarygradesshouldreadand haveonhand.Congratulations,friendAndy,foradifficulttaskwelldone! Norman,Oklahoma,USA, February,2009. ix