koncept of number system

8/10/2019 Koncept of Number System

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TOPICS

Numbers C lassi cation

Divisibility tests

Property of a Prime number

Property of perfect square numbers

Chapter

Numbers

Introduction2

Solved Examples : 25 Questions

Concept Applicator (CA): 20 Questions

Concept Builder (CB): 20 Questions

Concept Cracker (CC): 20 Questions

Concept Deviator (CD): 20 Questions

Data Suf ciency (DS) : 20 Questions

Total : 125 Questions


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NUMBERS

Numbers is one of the most important chapters for competitive exams . It is very interesting but can prove to be dif cult if the basic concepts are not understood clearly. There are numerous kinds of concepts and hencethe questions. In this chapter we will discuss some basic properties of numbers.

CLASSIFICATION OF NUMBERS

Numbers are broadly divided in twogroups Real numbers and Complexnumbers.

Complex Numbers : The numbers thatcannot be represented on the numberline are called Complex numbers. Thestandard from to represent a complexnumber is a + ib where a and b are realnumbers with b 0, andi = 1, herei is also called imaginary number. In acomplex numbera + ib , a is called real part whileb is called imaginary part. So acomplex number is a combination of realand imaginary number.

Properties of Imaginary number i :

We know thati = 1, hencei2

= 1,i3

= i, i4 = 1,i5 = 1,i6 = 1 so on ,itrepeats in cyclic manner with a cycle of4. Soi26 = ( i24)(i2) = 1.Conjugate of a Complex number : For a complex numbera + ib its conjugate is a ib. So to obtain theconjugate of a complex number what all we have to do is to change the sign of imaginary part.Sum of a complex number and its conjugate is always a real number.Real Numbers : All numbers that can be represented on the number line are called Real numbers. In thischapter we will mainly deal with Real numbers.All real numbers further can also be classi ed in two parts,Rational and Irrational numbers. Rational Numbers : All numbers that can be written in the form of p/q where q is not equal to zero is

called a rational numbers. Another de nition of rational number is all the terminating and recurringdecimals are rational numbers.Examples of rational numbers 5/2 , 22/3, 3, 22/7 etc.Rational numbers can be both positive as well as negative.

Irrational Number : All numbers that cannot be represented in the form of p/q where q is not equal tozero is called Irrational number. In other words all non recurring non terminating decimals are Irrationalnumber. e.g. .2, 3, 7,,e, etc.

CONVERSION OF RECURRI NG DECIMAL TO FRACTION:

All recurring numbers are rational numbers hence we can represent a recurring decimal to fraction.

Pure recurring decimals : These are the decimals where recurring starts just after the decimal,


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Example 1 : Convert 0.333333 in fraction.Solution :

Let x = 0.33333, (i)So 10 x = 3.333333. (ii)Subtract equation (i) from equation (ii) we will get

9 x = 3 or x =3 1

9 3=

Example 2 : Convert 0.67676767.. in fraction.Solution :

Let x = 0.67676767.. =0.67 (i)Or 100 x = 67.67 (ii)Subtracting equation (i) from equation (ii) we will get

99 x = 67, or x = 6799

Example 3 : Convert0.267 in fraction.Solution :

We know that 0.267 = 0.267267267267Let x = 0.267267267 (i)Or 1000 x = 267.267267267 (ii)Subtracting equation(i) from equation(ii) we will get

999 x = 267, x = 267 89999 333

=

Mixed recurring decimals : These are the types of decimals where recurring does not occur just after thedecimal but after appearance of certain number of digits for. e.g.0.267 , 0.2345 etcExample 4 : Convert0.267 in fraction.Solution :

Let x = 0.267 = 0.26767676767So 1000 x = 267.67676767 (i)And 10 x = 2.676767676767 (ii)subtracting equation(ii) from equation(i) we will get

990 x = 265, x = 265990

= 53198

Example 5 : Convert0.2345 in fraction.Solution :

Let x = 0.2345 or 0.234545454545So 10000 x = 2345.4545454545 (i)And 100 x = 23.454545 (ii)subtracting equation (ii) from equation (i) we will get.

9900 x = 2345 23 = 2322, x = 2322

9900
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So a pure recurring decimal can be converted to fraction by writing down the number which is recurring asnumerator and in denominator put 9s equal to the numbers on which recurring occurs.

So 0. ,0. ,0. 0.9 99 999 9999

= = = =a ab abc abcd

a ab abc abcd etc.

And to convert a mixed or impure recurring decimal into fraction, in the numerator write down the entirenumber formed by non recurring and recurring numbers and subtract from it the part of the decimal that isnot recurring. And in the denominator write as many 9s as the number of digits having recurring on it andthen next to it write as many zeroes as there are digits without recurring in the given decimal.

Hence 0.90

=ab a

ab , 0.990

=abc a

abc , 0.900

=abc ab

abc , 0.9900

=abcd ab

abcd

There are many ways to divide a group of real numbers, but for clear understanding, we will divide it in two parts Integers and Decimals .

Integers : Integers are again divided in three groups. Negative integers (e.g. 3, 5 etc) Zero (i.e. non negative and non positive integer) Positive integers (e.g. 3, 5 etc)Natural Numbers : All positive integers are called Natural Numbers. In other words all integers from 1 to are called Natural Numbers. So there are in nite number of natural numbers with the least natural number

is 1.Whole Numbers : When zero joins the group of natural number then the newly formed group is called Whole Number. (e.g. 0, 1, 2) Natural numbers can be divided in two groups EVEN and ODD Numbers.Even Numbers : All natural numbers that are divisible by 2 are called Even numbers. (e.g. 2, 4, 6 ).So all even numbers can be expressed in the form of 2n where n is a natural number.Odd Numbers : All natural numbers that are not divisible by 2 are called Odd numbers. In other words allnumbers that are in the form of 2 n + 1 wheren is a whole number or 2n 1 wheren is a natural number arecalled Odd numbers. (e.g. 1, 3, 5.)N.B : Negative integers can also be even numbers. i.e. 8, 6 are even integers.

Frac ons : It is divided in three broad categories Proper Fraction : The fractions in the form of

pq

, where p and q are integers such that q > p and q 0.

Improper Fraction : The fractions in the form of pq

, where p and q are integers such that q < p and q 0.

Mixed Fraction : Fraction in the form of a pq

where p and q are integers such that q > p, q 0 and a is aninteger.

N.B. : A proper fraction is less than 1 while an improper fraction is more than 1.Decimals : Decimals can be subdivided in three parts

Terminating decimals : A decimal number that terminate at a point, that means after that point only zerocomes. e.g. 4.25, 6.5432, 2.3. It has an end or terminating point hence it is called a terminating decimal.


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Recurring decimals : It is a type of decimal in which a digit or a set of digits is repeated continuously.Recurring decimals are written in abridged form, the digits that are repeated being marked by a bar placed

over it. e.g. 3.666666.. =3.6 , or 2.347347347 =2.347 . etc.

Non recurring non terminating decimals : Those decimals that does not form any pattern or are irrationalin pattern are called non recurring non terminating decimals.e.g.2, 3, 7,, e, etc.Proper es of Even and Odd Numbers

Odd + Odd = Even

Odd + Even = Odd

Even + Even = Even

Odd Odd = Odd

Odd Even = Even

Even Even = Even

Odd n = Odd (3 4 = 81, 33 = 27) Even n = Even (2 4 = 16, 25 = 32) here n is a natural number.

( )ve ven+ =+

( )Odd

ve ve =

( )Even

ve ve =+

SOME MORE PROPERTIES OF NATURAL NUMBER

Sum of 1st n Natural number i.e 1 + 2 + 3 + . +n = ( )12

n n +

Sum of squares of 1st n natural number i.e 12 + 2 2 + 3 2 + .. +n2 = ( )( )1 2 16

n n n+ +

Sum of cubes of 1st n Natural number i.e 13 + 2 3 + 3 3 + .. +n3 =( ) 21

2

n n +

Sum of squares of 1st n Odd natural number i.e 12 + 3 2 + 5 2 + .. + (2n 1)2 = ( )( )2 1 2 1

3

n n n +

Sum of squares of 1st n Even natural number i.e 22 + 4 2 + 6 2 + .. + (2n)2 = ( )( )2 1 2 13

n n n+ +

Sum of cubes of 1st n Odd natural number i.e ( ) ( )33 3 3 2 21 3 5 .. 2 1 2 1n n n+ + + + = Sum of cubes of 1st n Even natural number i.e ( ) 23 3 3 32 4 6 .. 2 2 1n n n+ + + + = +


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POWER SERIES

Summation Expansion Equivalent Value

1

n

k

k =

= 1 + 2 + 3 + 4 + +n = ( n2 + n)/2 = 2

1 1

2 2n n +

2

1

n

k

k =

12 + 2 2 + 3 2 + 4 2 + +n2 = (1/6)n(n + 1)(2n + 1) = 3 2

1 1 1

3 2 6n n n + +

3

1

n

k

k =

13 + 2 3 + 3 3 + 4 3 + +n3 = 4 3 21 1 1

4 2 4n n n + +

4

1

n

k

k =

14 + 2 4 + 3 4 + 4 4 + +n4 5 4 31 1 1 1 5 2 3 30n n n n = + +

5

1

n

k

k =

15 + 2 5 + 3 5 + 4 5 + +n5 6 5 4 21 1 5 1 6 2 12 12n n n n = + +

6

1

n

k

k =

16 + 2 6 + 3 6 + 4 6 + +n6 7 6 5 31 1 1 1 1 7 2 2 6 42n n n n n + + +

7

1

n

k

k =

17 + 2 7 + 3 7 + 4 7 + +n7 8 7 6 4 21 1 7 7 1 8 2 12 24 12n n n n n + + +

8

1

n

k

k =

18 + 2 8 + 3 8 + 4 8 + +n8 9 8 7 5 31 1 2 7 29 2 3 15 9 30n

n n n n n + + +

9

1

n

k

k =

19 + 2 9 + 3 9 + 4 9 + +n910 9

8 6 4 23 7 1 3 10 2 4 10 2 20

n nn n n n

+ + +

10

1

n

k

k =

110 + 2 10 + 3 10 + 4 10 + +n10 11 10 9 7 5 31 1 5 1 5 11 2 6 2 66n n n n n n n + + + +

VBODMAS

A given series of calculation or operation are solved with a speci c rule called VBODMAS. To understandthis let us see what these letters means

V Vinculum or Bar ( )B Brackets and the order of operation is ( ), { }, [ ],

O Of (Calculation is done same as multiplication) D Division

M MultiplicationA Addition

S Subtraction


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Example 1 : 90 (6 5) + (80/16 + 10 5)Solution :

90 30 + (5 + 10 5) = 90 30 + 10 = 70.Based on the divisibility, a group of natural numbers can be divided in two groupsPrime Numbers andComposite Numbers .

PRIME NUMBERS

Prime number is a natural number greater than unity that has only two factors the number itself and the unity.(e.g. 2, 3, 5, 7,.) The smallest prime number is 2. There is only one even prime number and that is 2. There are 15 prime numbers between 1 and 50. There are 25 prime numbers between 1 and 100. If p1 and p2 are two prime numbers > 2 then p12 p22 and p12 + p22 is composite number. All prime numbers > 3 can be represented in the form of 6n + 1 or 6n 1, but reverse of this is not always

true, that means all numbers in the form of 6n + 1 and 6n 1 may be or may not be a prime number. PROOF Let p is a prime number such that p > 3, Take 3 consecutive numbers ( p 1), p, and ( p + 1), since p is a prime number > 3 hence p has to be odd

and so ( p 1) and ( p + 1) has to be even.Out of three consecutive numbers one of them has to be divisible by 3, since p is a prime number it is not

divisible by 3. Let ( p 1) be divisible by 3 then since a number divisible by 3 and 2 (Even) then that number

has to be divisible by 6, so ( p 1) = 6n or p = 6 n + 1. But if ( p 1) is not divisible by 3 then ( p + 1) has to be divisible by 3 and it is even number hence it is divisible by 6 so p + 1 = 6n or p = 6 n 1. Hence a primenumber has to be in the form of 6 n + 1 or 6n 1.

( P 1) P ( P + 1) Even Number Prime Number Even Number Case (i) Divisible by 3 Not Divisible by 3 Not Divisible by 3 Divisible by 6 hence = 6n P = 6 n + 1Case (ii) Not Divisible by 3 Not Divisible by 3 Divisible by 3 P = 6 n 1 Divisible by 6 hence = 6n

Composite Number : A composite number is a number which has at least one factor other than 1 and thenumber itself. In other words any natural number other than 1 and prime number is a composite number. One

is neither a prime number nor a composite number.Relative Primes or Co-primes : A pair of numbers are said to be relative prime or co-prime to each other ifthey do not have any common factor other than 1.e.g 15 and 16. List down all the factors of 15 and 16, factorsof 15 = 1,3,5,15 and that of 16 = 1,2,4,8,16 so we can see that these two number have only one common factori.e. 1 hence these two are co-prime to each other. Two consecutive numbersn and n + 1 is always co-prime to each other. Two prime numbers are always co-prime to each other. Squares of two co-prime numbers are always co-prime to each other.Perfect Numbers : A natural number is said to be a perfect number if the sum of all its factors excluding thenumber itself but including 1 is equal to the number itself. e.g. 6 its factors are 1, 2 and 3 and sum of 1 + 2 +3 = 6 hence 6 is a perfect number. Take another example 8, its factors are 1, 2 and 4 and sum of its factors 1 +

2 + 4 = 7 and not equal to 8 hence it is not a perfect number. Other examples are 28, 496, 8128 etc.


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TO CHECK WHETHER A NUMBER IS PRIME OR NOT

To check whether a given number is prime or not 1st nd the square root of that number and then approximatethat to immediately lower integer (sayn) and write down all the prime numbers less than that integer. Thencheck the divisibility of the given number by all the prime numbers, if it is not divisible by any prime numbersthen given number is prime.To nd whether given number N is a prime or not.Step 1 : Find squre root of N say it as K ( Just nd approximate value) Let N = 211, approximate square root is 14.xxx. (Means more than 14 but less than 15)Step 2 : Write down all the prime numbers less than K All the prime numbers less than 14 are 2, 3, 5, 7, 11, and 13.Step 3 : Check divisibility of N with these prime numbers. Check divisibility of 211 by 2, 3, 5, 7,11 and 13, you will nd that it is not divisible by any of these

prime numbers, hence 211 is a prime number.Number of Digits Number of single digit natural numbers (i.e. from 1 9) = 9 Number of two digit natural numbers (i.e. from 10 99) = 90 Number of three digit natural number (i.e. from 100 999) = 900 etc.Lets consider any digit (Say 2)From 1 9 the digit 2 is used 1 timeFrom 1 99 the digit 2 is used 20 times.

From 1 999 the digit 2 is used 300 timesFrom 1 9999 the digit 2 is used 4000 times and so on

RULES FOR DIVISIBILITY

Rules for divisibility of some important numbers are as follows 2 All even number is divisible by 2, or the numbers end with and even number or zero is divisible by 2. 3 A number is divisible by 3 if sum of its digits is divisible by 3. 4 A number is divisible by if 4 if the number formed by the last two digits of the number is divisible

by 4, or last two digits are 0s. 5 All numbers that end with 0 or 5 are divisible by 5. 6 A number that is divisible by 2 and 3 both are divisible by 6.

8 A number divisible by 8 if number formed by three digits of the number is divisible by 8 or last threedigits are zeroes. 9 A number is divisible by 9 if sum of its digits is divisible by 9. 11 A number is divisible by 11 if the difference between the sum of the digits in the odd and the sum

of the digits in the even places is either 0 or multiple of 11. Example : Is 245718 divisible by 11? Solution : Here sum of the digits at odd places is 2 + 5 + 1 = 8, and sum of the digits at even places is

4 + 7 + 8 = 19 and the difference between these two is 19 8 = 11 that is divisible by 11 hence the givennumber is divisible by 11.

12 A number is divisible by 12 when it is divisible by 3 and 4 both. 16 A number is divisible by 16 when number formed by last 4 digits of the given number is divisible

by 16 or last 4 digits are zeroes.


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Observe the Pa ernHere if you see closely you will nd a relation Divisibility of 2, 2 = 21 so last 1 digit decides the divisibility of 2.Divisibility of 4, 4 = 22 so last 2 digit decides the divisibility of 4.Divisibility of 8, 8 = 23 so last 3 digit decides the divisibility of 8.Divisibility of 16, 16 = 24 so last 4 digit decides the divisibility of 16.So Divisibility of 32, 32 = 25 so last 5 digit decides the divisibility 32.

Explana on Why Above Rule Works Consider a four digit numberabcd and we have to check its divisibility by 4. We can writeabcd = ab

100 +cd now the 1st part i.e. ab 100 is always divisible by 4 so divisibility of the number by 4depends on last two digitscd .Similarly consider a ve digit numberabcde and we have to check its divisibility by 8. We can writeabcde = ab 1000 +cde , now again 1st part i.e. ab 1000 is always divisible by 8 so divisibility ofthe given number by 8 depends on last three digits.

Since 10 = 2 5 hence above property is true for both 2 and 5To check divisibility of 2 n or 5 n, just check the divisibility by last n digit.Hence if last 2 digit is divisible by 25, then number must be divisible by 25If last 3 digit is divisible by 125 then number must be divisible by 125, etc.

72 Now its your turn, how will you check the divisibility of a number by 72; if the number is divisible by 2 and 36, or if the number is divisible by 4 and 18 or the number is divisible by 8 and 9?

Look at the options one by one, is 2 and 36 correct. No not at all because if a number is divisible by36 then it has to be divisible by 2 so there is no meaning if I say that a number is divisible by 36 and 2

both. 2nd option 4 and 18 , that is also not the correct option because if a number is divisible by 18 then it is

divisible by 2 that is a factor of 4. Last option 8 and 9 is correct the reason is that these two numbers are co-prime to each other.

N.B So if a number n = a b where a and b are co-prime to each other, then a number is divisibleby n if the number is divisible by a and b both.

7 and 13 A number is divisible by 7 or 13 if and only if the difference of the number of its thousandsand the remainder of its division by 1000 is divisible by 7 or 13.

Example : Is 13153 divisible by 7.Solution : Here number of thousands = 13 and the remainder when it is divided by 1000 is 153 anddifference 153 13 = 140 is divisible by 7 hence the number 13153 is divisible by 7.

SOME IMPORTANT POINTS REGARDING DIVISIBILITY

When a number is divided byn, then there aren possible values of remainder from 0 ton 1. i.e. whena number is divided by 4 the possible value of the remainder is 0, 1, 2, & 3. And when remainder is 0 itis called completely divisible byn.

Out of a group ofn consecutive whole numbers one and only one number is divisible byn. Difference between any number (of two or more digits) and the number formed by reversing its digits

is always divisible by 9.


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The product of any n consecutive numbers is always divisible by n!

For any numbern (n p n ) is always divisible by p where p is a prime number. The square of an odd number when divided by 8 always leaves a remainder 1. For any natural number5n or 4 1k n + is having same unit digit as n has. Where k is a whole number.

Some Other Proper es of Numbers

Squares : There is a de nite relationship between the unit digits when square of a number is considered, wewill see that one by one. If unit digit of a perfect square is 1 then tens digit has to be Even (e.g. 81, 121, 441 etc)

If unit digit of a number is 2 then it cant be a perfect square. If unit digit of a number is 3 then it cant be a perfect square. If unit digit of a perfect square is 4 then tens digit has to be Even (e.g. 64, 144 etc) If unit digit of a perfect square is 5 then tens digit has to be 2 (e.g. 25, 225, 625 etc) If unit digit of a perfect square is 6 then tens digit has to be Odd and multiple of 3. If unit digit of a number is 7 & 8 then it cant be a perfect square If unit digit of a perfect square is 9 then tens digit has to be Even (e.g. 49, 169 etc) If unit digit of a perfect square is 0 then tens digit has to be 0 (e.g. 100, 400, 900 etc) Hence if a number end with 2, 3, 7 or 8 then it cant be perfect square.

SOME INTERESTING PROPERTIES OF NUMBERS

Beast Number

Consider a beast number 666 666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9 = 9 + 87 + 6 + 543 + 21666 is equal to the sum of the squares of the rst 7 primesi.e 2 2 2 2 2 2 22 3 5 7 11 13 17 666+ + + + + + =666 is a sum and difference of the 6 th powers of the rst three numbers.i.e 6 6 61 2 3 666+ =

Magical 1 :

3 37 = 111 33 3367 = 111,111 333 333667 = 111,111,111 3333 33336667 = 111,111,111,111 33333 3333366667 = 111,111,111,111,111 333333 333333666667 = 1111,111,111,111,111,111Consider LHS number of 3s in both the numbers are same and 6 is one less than the number of 3 and there is

only one 7. Then in the nal product N 1 would appear three times the number of threes.


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6 With 7 Give 4 and 2

6 7 = 42 66 67 = 4422 666 667 = 444222 6666 6667 = 44442222 66666 66667 = 4444422222 666666 666667 = 444444222222 6666666 6666667 = 44444442222222 66666666 66666667 = 4444444422222222 666666666 666666667 = 444444444222222222 Number of digits in both the numbers is same, the 2nd number has one 6 less than the number of 6s the 1st number has.

Number of 4s and 2s is equal to number of 6s in the 1st number.

Cyclic Number

142857 is called a cyclic number, since its digits are rotated around when multiplied by any number from 1to 6.

142857 1 = 142857142857 5 = 7 14285 142857 4 = 57 1428

142857 6 = 857 142 142857 2 = 2857 14 142857 3 = 42857 1

13 When Squared

132 = 1 6 9 [1, 6 and 9 only]1332 = 17 689 [insert one 7 between 1 & 6 and one 8 between 6 & 9] 13332 = 177 6889 [insert two 7s between 1 & 6 and two 8s between 6 & 9] 133332 = 1777 68889 [insert three 7s between 1 & 6 and three 8s between 6 & 9]1333332 = 17777 688889 [insert four 7s between 1 & 6 and four 8s between 6 & 9] 13333332 = 177777 6888889 [insert ve 7s between 1 & 6 and ve 8s between 6 & 9]


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koncept of number system

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