Study Materials for IBPS, Bank Exams _ Numerical Ability_Quantitative Aptitude - Number System _...
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Place Value (Indian) Face Value and Place Value of a Digit Face Value: It is the value of the digit itself eg, in 3452, face value of 4 is ‘four’, face value of 2 is ‘two’. Place Value: It is the face value of the digit multiplied by the place value at which it is situated eg, in 2586, place value of 5 is 5 × 102 = 500. Number Categories Natural Numbers (N): If N is the set of natural numbers, then we write N = {1, 2, 3, 4, 5, 6,…} The smallest natural number is 1. Whole Numbers (W): If W is the set of whole numbers, then we write W = {0, 1, 2, 3, 4, 5,…} The smallest whole number is 0. Integers (I): If I is the set of integers, then we write I = {– 3, –2, –1, 0, 1, 2, 3, …} Rational Numbers: Any number which can be expressed in the form of p/q, where p and q are both integers and q # 0 are called rational numbers. e.g. 3/2,7/9,5,2 There exists infinite number of rational numbers between any two rational numbers. Irrational Numbers Nonrecurring and nonterminating decimals are called irrational numbers. These numbers cannot be expressed in the form of p/q . e.g. √3, √5,√29 Real Numbers: Real number includes both rational and irrational numbers. Basic Rules on Natural Numbers 1. One digit numbers are from 1 to 9. There are 9 one digit numbers. ie, 9 × 10 . 2. Two digit numbers are from 10 to 99. There, are 90 two digit numbers. ie, 9 × 10. 3. Three digit numbers are from 100 to 199. There are 900 three digit numbers ie, 9 × 10 . In general the number of n digit numbers are 9 × 10 Sum of the first n, natural numbers ie, 1 + 2 + 3 + 4 + … + n = n n 1 / 2 Sum of the squares of the first n natural numbers ie. 1 +2 +3 +4 + …+ n = n n 1 2n 1 / 6 Different Types of Numbers Even Numbers: Numbers which are exactly divisible by 2 are called even numbers. eg, – 4, – 2, 0, 2, 4… 0 2 (n–1) 2 3 2 2 2
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Transcript of Study Materials for IBPS, Bank Exams _ Numerical Ability_Quantitative Aptitude - Number System _...
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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.
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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.
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