chapter 1: divisibility & primes

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