GRE Math Review 1 ArithmeticWord文档格式.docx
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TableofContents
OverviewoftheMathReview5
OverviewofthisChapter5
1.1Integers6
1.2Fractions11
1.3ExponentsandRoots16
1.4Decimals20
1.5RealNumbers24
1.6Ratio30
1.7Percent31
ArithmeticExercises39
AnswerstoArithmeticExercises44
OverviewoftheMathReview
TheMathReviewconsistsof4chapters:
Arithmetic,Algebra,Geometry,andDataAnalysis.
Eachofthe4chaptersintheMathReviewwillfamiliarizeyouwiththemathematicalskillsandconceptsthatareimportanttounderstandinordertosolveproblemsandreasonquantitativelyontheQuantitativeReasoningmeasureoftheGRE®
revisedGeneralTest.
ThematerialintheMathReviewincludesmanydefinitions,properties,andexamples,aswellasasetofexerciseswithanswersattheendofeachchapter.Note,however,thatthisreviewisnotintendedtobeallinclusive.Theremaybesomeconceptsonthetestthatarenotexplicitlypresentedinthisreview.Ifanytopicsinthisreviewseemespeciallyunfamiliarorarecoveredtoobriefly,weencourageyoutoconsultappropriatemathematicstextsforamoredetailedtreatment.
OverviewofthisChapter
ThisistheArithmeticChapteroftheMathReview.
Thereviewofarithmeticbeginswithintegers,fractions,anddecimalsandprogressestorealnumbers.Thebasicarithmeticoperationsofaddition,subtraction,multiplication,anddivisionarediscussed,alongwithexponentsandroots.Thechapterendswiththeconceptsofratioandpercent.
1.1Integers
Theintegersarethenumbers1,2,3,andsoon,togetherwiththeirnegatives,
negative1,negative2,negative3,dotdotdot,and0.
Thus,thesetofintegersis
dotdotdot,negative3,negative2,negative1,0,1,2,3,dotdotdot.
Thepositiveintegersaregreaterthan0,thenegativeintegersarelessthan0,and0isneitherpositivenornegative.Whenintegersareadded,subtracted,ormultiplied,theresultisalwaysaninteger;
divisionofintegersisaddressedbelow.Themanyelementarynumberfactsfortheseoperations,suchas
+
=
15,
78minus87=negative9,
7minusnegative18=25,and
7times8=56,
shouldbefamiliartoyou;
theyarenotreviewedhere.Herearethreegeneralfactsregardingmultiplicationofintegers.
Fact1:
Theproductoftwopositiveintegersisapositiveinteger.
Fact2:
Theproductoftwonegativeintegersisapositiveinteger.
Fact3:
Theproductofapositiveintegerandanegativeintegerisanegativeinteger.
Whenintegersaremultiplied,eachofthemultipliedintegersiscalledafactorordivisoroftheresultingproduct.Forexample,
2times3times10=60,
so2,3,and10arefactorsof60.Theintegers4,15,5,and12arealsofactorsof60,since
4times15equals60and5times12=60.
Thepositivefactorsof60are1,2,3,4,5,6,10,12,15,20,30,and60.Thenegativesoftheseintegersarealsofactorsof60,since,forexample,
negative2timesnegative30=60.
Therearenootherfactorsof60.Wesaythat60isamultipleofeachofitsfactorsandthat60isdivisiblebyeachofitsdivisors.Herearefivemoreexamplesoffactorsandmultiples.
ExampleA:
Thepositivefactorsof100are1,2,4,5,10,20,25,50,and100.
ExampleB:
25isamultipleofonlysixintegers:
1,5,25,andtheirnegatives.
ExampleC:
Thelistofpositivemultiplesof25hasnoend:
0,25,50,75,100,125,150,etc.;
likewise,everynonzerointegerhasinfinitelymanymultiples.
ExampleD:
1isafactorofeveryinteger;
1isnotamultipleofanyintegerexcept
1andnegative1.
ExampleE:
0isamultipleofeveryinteger;
0isnotafactorofanyintegerexcept0.
Theleastcommonmultipleoftwononzerointegersaandbistheleastpositiveintegerthatisamultipleofbothaandb.Forexample,theleastcommonmultipleof30and75is150.Thisisbecausethepositivemultiplesof30are30,60,90,120,150,180,210,240,270,300,etc.,andthepositivemultiplesof75are75,150,225,300,375,450,etc.Thus,thecommonpositivemultiplesof30and75are150,300,450,etc.,andtheleastoftheseis150.
Thegreatestcommondivisor(orgreatestcommonfactor)oftwononzerointegersaandbisthegreatestpositiveintegerthatisadivisorofbothaandb.Forexample,thegreatestcommondivisorof30and75is15.Thisisbecausethepositivedivisorsof30are1,2,3,5,6,10,15,and30,andthepositivedivisorsof75are1,3,5,15,25,and75.Thus,thecommonpositivedivisorsof30and75are1,3,5,and15,andthegreatestoftheseis15.
Whenanintegeraisdividedbyanintegerb,wherebisadivisorofa,theresultisalwaysadivisorofa.Forexample,when60isdividedby6(oneofitsdivisors),theresultis10,whichisanotherdivisorof60.Ifbisnotadivisorofa,thentheresultcanbeviewedinthreedifferentways.Theresultcanbeviewedasafractionorasadecimal,bothofwhicharediscussedlater,ortheresultcanbeviewedasaquotientwitharemainder,wherebothareintegers.Eachviewisuseful,dependingonthecontext.Fractionsanddecimalsareusefulwhentheresultmustbeviewedasasinglenumber,whilequotientswithremaindersareusefulfordescribingtheresultintermsofintegersonly.
Regardingquotientswithremainders,considertwopositiveintegersaandbforwhichbisnotadivisorofa;
forexample,theintegers19and7.When19isdividedby7,theresultisgreaterthan2,since
2times7islessthan19,butlessthan3,since
19islessthan3times7.Because19is5morethan
2times7,wesaythattheresultof19dividedby7isthequotient2withremainder5,orsimply2remainder5.Ingeneral,whenapositiveintegeraisdividedbyapositiveintegerb,youfirstfindthegreatestmultipleofbthatislessthanorequaltoa.Thatmultipleofbcanbeexpressedastheproductqb,whereqisthequotient.Thentheremainderisequaltoaminusthatmultipleofb,or
r
aminusqb,whereristheremainder.Theremainderisalwaysgreaterthanorequalto0andlessthanb.
Herearethreeexamplesthatillustrateafewdifferentcasesofdivisionresultinginaquotientandremainder.
100dividedby45is2remainder10,sincethegreatestmultipleof45that’slessthanorequalto100is
2times45,or90,whichis10lessthan100.
24dividedby4is6remainder0,sincethegreatestmultipleof4that’slessthanorequalto24is24itself,whichis0lessthan24.Ingeneral,theremainderis0ifandonlyifaisdivisiblebyb.
6dividedby24is0remainder6