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PlaceValue(Indian)

FaceValueandPlaceValueofaDigitFaceValue:Itisthevalueofthedigititselfeg,in3452,facevalueof4isfour,facevalueof2istwo.PlaceValue:Itisthefacevalueofthedigitmultipliedbytheplacevalueatwhichitissituatedeg,in2586,placevalueof5is5102=500.

NumberCategoriesNaturalNumbers(N):IfNisthesetofnaturalnumbers,thenwewriteN={1,2,3,4,5,6,}Thesmallestnaturalnumberis1.WholeNumbers(W):IfWisthesetofwholenumbers,thenwewriteW={0,1,2,3,4,5,}Thesmallestwholenumberis0.

Integers(I):IfIisthesetofintegers,thenwewriteI={3,2,1,0,1,2,3,}RationalNumbers:Anynumberwhichcanbeexpressedintheformofp/q,wherepandqarebothintegersandq#0arecalledrationalnumbers.

e.g.3/2,7/9,5,2

Thereexistsinfinitenumberofrationalnumbersbetweenanytworationalnumbers.IrrationalNumbersNonrecurringandnonterminatingdecimalsarecalledirrationalnumbers.Thesenumberscannotbeexpressedintheformofp/q.

e.g.3,5,29

RealNumbers:Realnumberincludesbothrationalandirrationalnumbers.

BasicRulesonNaturalNumbers1.Onedigitnumbersarefrom1to9.Thereare9onedigitnumbers.ie,910 .2.Twodigitnumbersarefrom10to99.There,are90twodigitnumbers.ie,910.3.Threedigitnumbersarefrom100to199.Thereare900threedigitnumbersie,910 .Ingeneralthenumberofndigitnumbersare910

Sumofthefirstn,naturalnumbersie,1+2+3+4++n=nn1/2

Sumofthesquaresofthefirstnnaturalnumbersie.1 +2 +3 +4 ++n =nn12n1/6

DifferentTypesofNumbersEvenNumbers:Numberswhichareexactlydivisibleby2arecalledevennumbers.eg,4,2,0,2,4

0

2

(n1)

2 3 2 2 2

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Sumoffirstnevennumbers=n(n+1)OddNumbers:Numberswhicharenotexactlydivisibleby2arecalledoddnumbers.eg,5,3,1,0,1,3,5Sumoffirstnoddnumbers=nPrimeNumbers:Numberswhicharedivisiblebyoneanditselfonlyarecalledprimenumbers.eg,2,3,5,7,11

2istheonlyevenprimenumber.

1isnotaprimenumberbecauseithastwoequalfactors.

Everyprimenumbergreaterthan3canbewrittenintheformof(6K+1)or(6K1)whereKisaninteger.

Thereare15primenumbersbetween1and50andl0primenumbersbetween50and100.

RelativePrimeNumbers:Twonumbersaresaidtoberelativelyprimeiftheydonothaveanycommonfactorotherthan1.eg,(3,5),(4,7),(11,15),(15,4)TwinPrimes:Twoprimenumberswhichdifferby2arecalledtwinprimes.eg,(3,5),(5,7),(11,13),CompositeNumbersNumberswhicharenotprimearccalledcompositenumberseg,4,6,9,15,1isneitherprimenorcomposite.PerfectNumber:Anumberissaidtobeaperfectnumber,ifthesumofallitsfactorsexcludingitselfisequaltothenumberitself.eg,Factorsof6are1,2,3and6.Sumoffactorsexcluding6=1+2+3=6.6isaperfectnumber.Otherexamplesofperfectnumbersare28,496,8128etc.

RulesforDivisibility

Divisibilityby2:Anumberisdivisibleby2whenthedigitatonesplaceis0,2,4,6or8.eg,3582,460,28,352,....

Divisibilityby3:Anumberisdivisibleby3whensumofalldigitsofanumberisamultipleof3.eg,453=4+5+3=12.12isdivisibleby3so,453isalsodivisibleby3.

Divisibilityby4:Anumberisdivisibleby4,ifthenumberformedwithitslasttwodigitsisdivisibleby4.eg,ifwetakethenumber45024,thelasttwodigitsform24.Since,thenumber24isdivisibleby4,thenumber45024isalsodivisibleby4.

Divisibilityby5:Anumberisdivisibleby5ifitslastdigitis0or5.eg,10,25,60

Divisibilityby6:Anumberisdivisibleby6,ifitisdivisiblebothby2and3.eg,48,24,108

Divisibilityby7:Anumberisdivisibleby7whenthedifferencebetweentwicethedigitatonesplaceandthenumberformedbyotherdigitsiseitherzerooramultipleof7.eg,6586528=6516=49As49isdivisibleby7thenumber658isalsodivisibleby7.

2

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Divisibilityby8:Anumberisdivisibleby8,ifthenumberformedbythelast3digitsofthenumberisdivisibleby8.eg,ifwetakethenumber57832,thelastthreedigitsform832.Since,thenumber832isdivisibleby8,thenumber57832isalsodivisibleby8..

Divisibilityby9:Anumberisdivisibleby9,ifthesumofallthedigitsofanumberisamultipleof9.eg,684=6+8+4=18.18isdivisibleby9so,684isalsodivisibleby9.

Divisibilityby10:Anumberisdivisibleby10,ifitslastdigitis0.eg,20,180,350,.

Divisibilityby11:Whenthedifferencebetweenthesumofitsdigitsinoddplacesandinevenplacesiseither0oramultipleof11.

eg,304263+4+6=130+2=2132=11

Asthedifferenceisamultipleof11thenumber30426isalsodivisibleby11.

SmartFacts

IfpandqarecoprimesandbotharefactorsofanumberK,thentheirproductpxqwillalsobeafactorofr.eg,Factorsof24are1,2,3,4,6,8,12and24primefactorsof24are2and3,whicharecoprimealso.Productof23=6,6isalsoafactorof24.

Ifpdividesqandr,thenpalsodividestheirsumordifference.eg,4divides12and20.Sumof12and20is32whichisdivisibleby4.Differenceof20and12is8whichisdivisibleby4.

Ifanumberisdivisiblebyanothernumber,thenitmustbedivisiblebyeachofthefactorsofthatnumber.48isdivisibleby12.Factorsof12are1,2,3,4,6,12.So,48isdivisibleby2,3,4and6also.

DivisiononNumbersInasumofdivision,wehavefourquantities.Theyare(i)Dividend,(ii)Divisor,(iii)Quotientand(iv)Remainder.Thesequantitiesareconnectedbyarelation.(a)Dividend=DivisorQuotient+Remainder.(b)Divisor=(DividendRemainder)Quotient.(c)Quotient=(DividendRemainder)Divisor.

Example2:Inasumofdivision,thequotientis110,theremainderis250,thedivisorisequaltothesumofthequotientandremainder.Whatisthedividend?Solution.Divisor=(110+250)=360Dividend=(360110)+250=39850Hence,thedividendis39850.

Example3:Findthenumberofnumbersupto600whicharedivisibleby14.Solution.Divide600by13,thequotientobtainedis46.Thus,thereare46numberslessthan600whicharedivisibleby14.

FactorsandMultiples

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Factor:Anumberwhichdividesagivennumberexactlyiscalledafactorofthegivennumber,eg,24=124,212,38,46

Thus,1,2,3,4,6,8,12and24arefactorsof24.1isafactorofeverynumberAnumberisafactorofitselfThesmallestfactorofagivennumberis1andthegreatestfactoristhenumberitself.Ifanumberisdividedbyanyofitsfactors,theremainderisalwayszero.Everyfactorofanumberiseitherlessthanoratthemostequaltothegivennumber.Numberoffactorsofanumberarefinite.

NumberofFactorsofaNumber:IfNisacompositenumbersuchthatN=a b c ...wherea,b,c...areprimefactorsofNandm,n,o...arepositiveintegers,thenthenumberoffactorsofNisgivenbytheexpression(m+1)(n+1)(o+1)

Example4:Findthenumberoffactorsthat224has.Solution.224=2571Hence,224has(5+1)(1+1)=62=12factors.

Multiple:Amultipleofanumberisanumberobtainedbymultiplyingitbyanaturalnumbereg,Multiplesof5are5,10,15,20Multiplesof12are12,24,36,48Everynumberisamultipleof1.Thesmallestmultipleofanumberisthenumberitself.Wecannotfindthegreatestmultipleofanumber.Numberofmultiplesofanumberareinfinite.

m n o