chapter 1: divisibility & primes
TRANSCRIPT
Chapter1:Divisibility&PrimesAnintegerisdivisibleby:2 Iftheintegeriseven. 3 IftheSUMoftheinteger's
DIGITSisdivisibleby3.4 IftheLASTTWOdigitsare
divisibleby4.5 iftheintegerendsin0or5. 6 iftheintegerisdivisibleby
BOTH2and3.7 PerformLongDivision
8 iftheintegerisdivisibleby2THREETIMFS,oriftheLASTTHREEdigitsaredivisibleby8.
9 iftheSUMoftheinteger'sDIGITSisdivisibleby9.
10 iftheintegerendsin0.
• FewerFactors,moremultiples:Anyintegeronlyhasalimitednumberoffactors.Bycontrast,thereisaninfinitenumberofmultiplesofaninteger.
• AneasywaytofindallthefactorsofaSMALLnumberistousefactorpairs• Oneveryhelpfulwaytoanalyzeanumberistobreakitdownintoitsprimefactors.
(1) Determiningwhetheronenumberisdivisiblebyanothernumber(2) Determiningthegreatestcommonfactoroftwonumbers(3) Reducingfractions(4) Findingtheleastcommonmultipleoftwo(ormore)numbers(5) Simplifyingsquareroots(6) Determiningtheexponentononesideofanequationwithintegerconstraints
• GreatestCommonFactor(GCF):thelargestdivisoroftwoormoreintegers• LeastCommonMultiple(LCM):thesmallestmultipleoftwoormoreintegers.
• àDonotdoublecount.
Chapter2:Odds&Evens• IfthereareXevenintegersinasetofintegersbeingmultipliedtogether,theresultwillbedivisibleby2X.• Thesumofanytwoprimeswillbeeven,unlessoneofthoseprimesisthenumber2.
Chapter3:Positives&NegativesStrategies
Inthesesituations,youshouldsetupatablelistingallthepossiblepositive/negativecombinationsofthevariables.
Chapter4:ConsecutiveIntegers
• {12,16,20,24}isasetofconsecutivemultiples Average=Sum/Number• PropertiesofEvenlySpacedSets
• Thearithmeticmean(average)andmedianareequaltoeachother.• ThemeanandmedianofthesetareequaltotheaverageoftheFIRSTandLASTterms.
• Rememberthatifbothextremesshouldbecounted,youneedtoaddonebeforeyouaredone.§ Ex:Howmanyintegersaretherefrom14to765,inclusive?
• ConsecutiveMultiples:((Last–First)/Increment)+1§ Ex:Alloftheevenintegersbetween12and24
• TheSumofConsecutiveIntegers:• Usethemethodsabove:Findthemedian/Averageandmultiplybythenumberofintegers.
§ Ex:Whatisthesumofalltheintegersfrom20to100,inclusive?
• Foranyoddnumberofconsecutiveintegers,thesumofthoseintegersisdivisiblebythenumberofintegers.§ Thismeansthattheaverageisaninteger.Onlyoddconsecutivenumbershaveanintegerasaverage.
Thus,kisodd.• ProductsofConsecutiveIntegersandDivisibility
• Theproductofkconsecutiveintegersisalwaysdivisiblebykfactorial(k!).• SumsofConsecutiveIntegersandDivisibility
• ForanysetofconsecutiveintegerswithanODDnumberofitems,thesumofalltheintegersisALWAYSamultipleofthenumberofitems.(Ex:4+5+6+7+8=30)
• ForanysetofconsecutiveintegerswithanEVENnumberofitems,thesumofalltheitemsisNEVERamultipleofthenumberofitems.(Ex:1+2+3+4=10)
• ConsecutiveIntegersandDivisibility:§ Ifxisaneveninteger,isx(x+l)(x+2)divisibleby4?
Useaprimeboxtokeeptrackoffactorsofconsecutiveintegers.(xiseven,soitisdivisibleby2).(x+2Iseven,soitisdivisibleby2).Theproductisdivisibleby2x2=4
Chapter5:Exponents
• Youcanonlysimplifyexponentialexpressionsthatarelinkedbymultiplicationordivisionififtheyhaveeitherabaseoranexponentincommon.Youcannotsimplifyexpressionslinkedbyadditionorsubtraction(althoughinsomecases,youcanfactorthemandotherwisemanipulatethem).
• Fewproblems:
Chapter6:Roots(alsocalledradicals)• Rule:Evenrootsonlyhaveapositivevalue.SQRT(4)=2,NOT±2.• Withintheexponentfraction,thenumeratortellsuswhatpowertoraisethebaseto,
andthedenominatortellsuswhichroottotake.
• 216=2x2x2x3x3x3=63• Rule:Youcanonlysimplifyrootsbycombiningorseparatingtheminmultiplicationand
division.Youcannotcombineorseparaterootsinadditionorsubtraction.
• ImperfectSquares:Wecanrewriteimperfectsquaresasaproductofprimesunderthe
radical.
Chapter7:PEMDAS
• Thecorrectorderofoperationsis:Parentheses-Exponents-(Multiplication-Division)-(Addition-Subtraction).• PleaseExcuseMyDearAuntSally.• Payattentiontox–(y–z).Distribute.
Chapter8:StrategiesforDataSufficiency
• Yourfirsttaskinsolvingadatasufficiencyproblemistorephrasethequestionand/orthestatementswheneverpossible.Afterrephrasingthequestion,youshouldalsotrytorephraseeachofthetwostatements,ifpossible.
àAretheretwo3'sanda2intheprimeboxofp
àisxeven?• TypesofDataSufficiencyProblems:Valuevs.Yes/No
àTestNumbers:n=1,2,3,or4• TestSmartNumbers:tryyourbesttofindnumbersthatyieldmultipleanswersforaValuequestion,oraMAYBE
answerforaYES/NOquestion.
• Wheneveryoufindthatyourtwostatementscontradicteachother,itmeansthatyouhavemadeamistake.
Chapter10:DIVISIBILITY&PRIMES:ADVANCED
• Alltheprimesupto100(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97)• GCFandLCMfor3numbers(ormore):
• ThreegeneralpropertiesoftheGCFandLCMareworthnoting:
- (GCFofmandn)x(LCMofmandn)=mxn- TheGCFofmandncannotbelargerthanthedifferencebetweenmandn.- ConsecutivemultiplesofnhaveaGCFofn.
àSincezisnotdivisibleby2,itcannotbedivisibleby6.
àDoNOTdoublecount
àUsethetableabove
• Perfectsquaresalwayshaveanoddnumberoffactors;Otherintegersalwayshaveanevennumberoffactors
• Perfectsquaresareformedfromtheproductoftwocopiesofthesameprimefactors.Therefore,theprimefactorizationofaperfectsquarecontainsonlyevenpowersofprimes.
• PrimefactorsofperfectsquaresMUSTcomeinpairs;likewise,primefactorsofperfectcubesMUSTcomeingroupsof3.
àDotheprimefactorizationof240andk3
• N!istheproductofallpositiveintegerssmallerthanorequaltoN.Therefore,N!mustbedivisiblebyallintegersfrom1toN.
•
àb>=6,d>=9• Twousefultipsforarithmeticwithremainders,ifyouhavethesamedivisorthroughout:
(1) Youcanaddandsubtractremaindersdirectly,aslongasyoucorrectexcessornegativeremainders."Excessremainders"areremainderslargerthanorequaltothedivisor.Tocorrectexcessornegativeremainders,justaddorsubtractthedivisor.
(2) Youcanmultiplyremainders,aslongasyoucorrectexcessremaindersattheend• 17/5=3.4.Thisquotienthasanintegerportion(3)andadecimalportion(0.4).Thedecimalportionrepresents
theremainder2dividedby5.
àR/B=7/20,7B=2*2*5*R• CountingTotalFactors:
- Howmanydifferentfactorsdoes2,000have?§ First,factor2,000intoprimes:2,000=24X53.
Sothetotalnumberoffactorsof2,000mustbe(4+1)(3+1)=5x4=20differentfactors.- Ifanumberhasprimefactorizationaxxbyxcz(wherea,b,andcareallprime),thenthenumberhas(x+1)(y
+1)(z+1)differentfactors.• InterestingProblems:
Chapter11:ODDS&EVENS/POSITIVES&NEGATIVES/CONSECUTIVEINTEGERS:ADVANCED• SpecialCaseofDivisibility(Odds&Evens)
àUsePrimeBox• Divisibilityby2hasaspecialpropertythatdivisibilitybyothernumbersdoesnothave.RecallfromChapter10
thatingeneralwhenweaddorsubtracttwonumbers,neitherofwhichisdivisiblebyx,wecannottellwhethertheresultwillbedivisiblebyx.However,whenaddingorsubtractingtwointegers,neitherofwhichisdivisibleby2,theresultwillalwaysbedivisibleby2.
• RemainderRulestoRemember:- Oddintegersarethoseintegersthatleavearemainderof1afterdivisionby2.- Evenintegersarethoseintegersthatleavearemainderof0afterdivisionby2.
• RepresentingEven&Oddalgebraically:EvenInteger:2n OddInteger:2n+1or2n-1
à4n2+4n+1.Multiplesof4haveremainder0• AbsoluteValueofaDifference:48<x<54canberewritten|x-51|<3
• DisguisedPositive,&.NegativeQuestions:
à(2)doesnottellusifa>b;(2)isinsuficientGenerallyspeaking,wheneveryouseeinequalitieswithzerooneithersideoftheinequality,youshouldconsidertestingpositive/negativecasestohelpsolvetheptoblem.
• ComplexAbsoluteValueEquations:WithanabsolutevalueequationthatcontainsmorethanonevariableandNOconstants,itisusuallyeasiesttotestpositive/negativenumberstosolvetheproblem.
Notethat|x|hastobebiggerthanorequalto|y|,since|x|-|y|isequaltoanabsolutevalueand|x|-|y|>=0
àcriterion:differentsign
• ConsecutiveIntegersandDivisibility:
àp=(x-1)(x)(x+1)andcisodd(x-1)and(x+1)areconsecutivemultiplesof2.Soeither(x-1)or(x+1)musthaveanother2andbedivisibleby4.Therefore,Pisdivisibleby8.Inaddition,oneofthenumbers–(x-1),x,or(x+1)–isdivisibleby3,becauseinanysetof3consecutiveinte-gers,oneoftheintegerswillbeamultipleof3.Wecanthereforeconcludethatifxisodd,Pwillbedivisiblebyatleast2x2x2x3=24.
Chapter12:EXPONENTS&ROOTS:ADVANCED
• SimplifyingExponentialExpressions:
4StepsProcess:
(1) Simplifyoffactoranyadditiveorsubstractiveterms(2) Breakeverynon-primebaseintoprimefactors(3) Distributetheexponentstoeveryprimefactor(4) Combinetheexponentsforeachprimefactorandsimplify
Ageneralruleofthumbisthatwhenyouencounteranyexponentialexpressioninwhichtwoormoretermsincludesomethingcommoninthebase,youshouldcomiderfactoring.Similarly,whenanexpressionisgiveninfactoredform,conslderdistributingit.
• SimplifyingRootswithPrimeFactorization:
Ex:SQRT(180)=SQRT(3*3*2*2*5)Tosimplifyaroot,followthisprocedure:
(1) Factorthenumberundertheradicalsignintoprimes.(2) Pulloutanypairofmatchingprimesfromunderthe
radicalsign,andplaceoneofthoseprimesoutsidetheroot.
(3) Consolidatetheexpression.
• AddingandSubtractingRoots:Rootsactlikevariablesinadditionandsubtraction:youcanonlycombinethemiftheyare"liketerms"orsimilarterms.
à=SQRT(5)Therefore,youmustsimplifyrootsbeforeyouaddorsubtractthemtoseewhetherthefinalnumberundertheradicalisthesame.Sometimerootsthatdonotappearatfirsttobesimilarcaninfactbecombined.
• UsingConjugatestoRationalizeDenominators