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Richard S. Millman · Peter J. Shiue Eric Brendan Kahn Problems and Proofs in Numbers and Algebra Problems and Proofs in Numbers and Algebra Richard S. Millman • Peter J. Shiue Eric Brendan Kahn Problems and Proofs in Numbers and Algebra 123 RichardS.Millman PeterJ.Shiue SchoolofMathematics DepartmentofMathematicalSciences GeorgiaInstituteofTechnology UniversityofNevada,LasVegas Atlanta,GA,USA LasVegas,NV,USA EricBrendanKahn DepartmentofMathematics, ComputerScienceandStatistics BloomsburgUniversity Bloomsburg,PA,USA ISBN978-3-319-14426-9 ISBN978-3-319-14427-6 (eBook) DOI10.1007/978-3-319-14427-6 LibraryofCongressControlNumber:2014960209 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface The transition from studying calculus or differential equations to learning about proofsis one thatis enormouslyinterestingas it showshowexcitingmathematics canbe.Themostimportantaspectofthisstageofthetransitionforstudentsisthe needforrigorousmathematicalreasoning.Thebenefitstoreaderswhoaremoving fromcalculustomoreabstractmathematicsaretodeveloptheabilitytounderstand proofsthroughProblemsandProofsinNumbersandAlgebra(PPNA)ontheirways toanalysis,abstractalgebra,etc.whichcomenext.Ourgoalisforstudentstofocus onhowtobothprovetheoremsandsolveproblemsetswhichhavedepth—multiple stepsareneededtoproveorsolve. Our approach,“solving rigorousproblemsand learning how to prove,” gives a platformoftwospecificcontentthemes,numbertheoryandalgebra(polynomials), with some aspects of applications. Undergraduatemathematics students will then learnhowto proveandsolveproblemsbecausetheyarecomfortablewith thetwo content themes. Furthermore, our approach is that the content areas of this book allowstudentstodevelopanaturalandconceptualunderstandingofthemathematics on their path forward. Students will study the conceptwith clarity, precision, and a mathematical habit of the mind through these problems. They will gain the foundationsofmathematicalprooftechniquesandstyles. Thekeytothetextisitsinterestingandintriguingproblems,exercises,theorems, andproofs,showinghowstudentswilltransitionfromtheusual,moreroutinecalcu- lustoabstractionwhilealsolearninghowto“prove”or“solve”.Itsapplicationssuch asRSAcryptosystems,UniversalProductCode(UPC),andInternationalStandard BookNumber(ISBN)areincludedinsectionsofthethirdchapter. Problems and proofs are the heart of mathematics. The goal of conceptual understanding grows as a large number of problems and examples that reward curiosityandinsightfulnessoversimplicity.Theproblemsaremulti-stepandrequire the reader to think. An intriguing variety of problems range from moderate to thoughtfultodeep. Eachproblemsetbeginswithafeweasyproblems.Afterthat,andincoordina- tionwithourapproachtothesubjectofdepthandconceptualunderstanding,many of the other problems require multi-step solutions, whether they be problems or v vi Preface proofs.Inaddition,thereareexercisesinthetext;thisdifferencebetweenexercises, examples, and problem sets is that the exercises stay close to the examples of the section allowing students the immediate opportunity to practice developing techniques. Furthermore, some problems are motivated from various mathematics competitions. Dr. Shiue’s significant experience with problems includes constructing problemsand proofswhile he is involved in the American Regional Mathematics League (1997–2009), American Mathematical Competitions/MAA (since 1998), TaiwanRegionalMathematicsLeague(since1997),andTaiwanJuniorHighSchool MathematicsCompetitions(since2002).Hisworkwithcompetitionshasenriched thequalityofourproblems. Someoftheconceptsincludethefollowing: NumberTheory,Algebra,Proofswithapproachestothem,DivisionAlgorithm,Euclidean Algorithm,GreatestCommonDivisor,LeastCommonMultiple,theRemainderTheorem, Diophantine Equations and Counting, Equivalence Classes, Divisibility of both Integers andPolynomials,FactoringPolynomialsandRoots,Matrices(intheplaneandin3-space), Cramer’sRule,andDeterminants,amongothers. This text has been revised by the authors over 7 years. An earlier course has beenusedtwiceattheUniversityofKentucky(ProblemSolvingforMiddleSchool Teachers) and at the University of Nevada, Las Vegas (four times as a secondary resource for future teachers). High school teachers can use the PPNA material in theirclassroomforstrongandadvancedstudentsthroughnumbersandalgebra. An advanced math course, Problems and Proof, at the public high school in Georgia, Gwinnett School of Math, Science and Technology (GSMST), has been a basis for PPNA. PPNA has been revised each spring semester from 2011 to 2014 by the authors and four Georgia Institute of Technology graduate students. Justin Boone has done very well as the individual who not only has helped with the typesetting via LATEX but also has given us fine advice. We very much appreciatetheadviceofDanielConnelly,S.GreysonDaugherty,andNolanLeung fortheirexcellencein teachingthe PPNA courseatGSMST andthe helpofScott MacDonald,graduatestudentatUNLV. Wethankthehighschoolandcollegestudentsandteacherswhoworkedthrough the variousrevisionsof the draft,who were a pleasureto collaboratewith us, and whoaretocontinuetomakemathematicsanevenmoreinterestingplace.Wewould behappytoreceivecommentsaboutthebookandtorespond.Pleasesendtorichard. [email protected],[email protected],[email protected]. Dr.Millman’sgranddaughter,BlumaMillman,enjoysawonderfulmathematics majorwithalgebraandnumbertheoryandSandyhashelpedmuch.Dr.Shiuewould liketothankhiswife,Stella,forherfullsupportduringthepreparationofthisbook, and it is with greatpleasure that Dr. Kahn would like to expresshis fullgratitude tohiswifeEmilyforhersupportandencouragementthroughoutallphasesofthis project. Preface vii In addition, Dr. Tian-Xiao He, Illinois Wesleyan University, and Dr. William Speer, UNLV, have reviewed PPNA much and given us good advice. We thank Dr.DerrickDuBose,Chairman,DepartmentofMathematicalSciences,UNLV,for hissupport. Atlanta,GA RichardS.Millman LasVegas,NV PeterJ.Shiue Bloomsburg,PA EricBrendanKahn Contents PartI TheIntegers 1 NumberConcepts,PrimeNumbers,andtheDivisionAlgorithm...... 3 1.1 BeginningNumberConceptsandPrimeNumbers.................... 5 1.2 DivisibilityofSomeCombinationsofIntegers........................ 12 1.3 LongDivision:TheDivisionAlgorithm............................... 17 1.4 TestsforDivisibilityinBaseTen....................................... 22 1.5 BinaryandOtherNumberSystems .................................... 31 1.5.1 ConversionBetweenBinaryandDecimal..................... 33 1.5.2 ConversionfromDecimaltoBinary........................... 33 1.5.3 ArithmeticinBinarySystems.................................. 34 1.5.4 DuodecimalNumberSystem................................... 37 2 GreatestCommonDivisors,DiophantineEquations,and Combinatorics................................................................ 41 2.1 GCDandLCMThroughtheFundamentalTheoremofArithmetic.. 41 2.2 GCD,theEuclideanAlgorithmandItsByproducts................... 51 2.3 LinearEquationswithIntegerSolutions:DiophantineEquations.... 61 2.4 ABriefIntroductiontoCombinatorics................................. 69 2.5 LinearDiophantineEquationsandCounting.......................... 75 3 EquivalenceClasseswithApplicationstoClockArithmetic andFractions................................................................. 79 3.1 EquivalenceRelationsandEquivalenceClasses ...................... 79 3.2 Modular(Clock)ArithmeticThroughEquivalenceRelations........ 87 3.3 FractionsThroughEquivalenceRelations............................. 94 3.4 IntegersModularnandApplications.................................. 101 3.4.1 RSACryptosystem ............................................. 102 3.4.2 UPCandISBN(SeeGallinandWinters[3],Rosen[10]).... 105 ix