Quantitative Aptitude

                                1.Number System

• Numbers are the collection of certain symbols or figures called digits
• The common number system in use is the decimal number system.
• In this system we use digits as 0,1,2,3,4,5,6,7,8 and 9.

        A combination of these digits representing a number is called a Numeral.

FACE VALUE AND PLACE VALUE

• Face value is equal to the value of the digit itself.

Example face value of 3 in 47326          = 3

Face value of 7 in 47326                          = 7

• Place value is equal to the place of the given digit .

We begin from the extreme right as unit’s place, ten’s place, hundred’s place, thousand’s place and so on.

Place value of 5 = 5 x 100 =      500

Place value of 1 = 1×10             =  10

Eg: 276345

TYPES OF NUMBERS

1. Natural Numbers (N)

Counting numbers such as  1,2,3,4………… are called natural numbers.

Eg:  N =1,2,3,4,……. ()infinity

2. Whole Numbers (W)

Includes all natural numbers and zero.

Eg.  W = 0,1,2,3,……. ()infinity

3. Even Natural Numbers

These are the numbers which are completely divisible by 2.

Eg:  2,4,6,8……………

4. Odd Natural Numbers

Numbers which are not divisible by 2.

Eg:  1,3,5,7,………………

Note:  Zero is an exception to the even – odd classification.

5. Integers     (I)

    Includes  all whole numbers along with negative numbers.

    Eg:   I …….-2,-1,0,1,2,…………. .(infinity)

Positive Integers

This includes all natural numbers.

Eg:  1,2,3………. (infinity)

Negative Integers

  This includes -1,-2……….. (infinity)

  Note:    Zero is neither positive nor negative.

6. Rational Numbers:

Real numbers which are expressed in the form of fractions like ‘a/b ‘ where a and b are integers and b not equal to Zero are called rational numbers.

Eg:  3/7, 8/5 , 3 (3/1), -9 (-9/1)

7. Irrational numbers 

Real numbers which cannot be expressed in the form of fractions like a/b, where b not equal to 0 are called irrational numbers.

Eg:  7,3,√5

8. Prime Numbers

Numbers with only two factors   → 1 and that number itself.

Eg:  2,3,5,7,11

2 = 1×2

3 =1×3

5 =1×5

  CODE

Prime numbers between 1 and 100

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

Prime Number

9 .Composite Numbers

Numbers with more than 2 factors.

Eg:  4,6,8,9,10,12,14,……………..

 4   = 1 x 4               3 factors = 1,2,4

      =  2 x 2

9    = 1 x 9

      = 3 x 3

10. Square Numbers

1,4,9,16,25,36

1²  = 1

2²  = 4

11. Cube Numbers

1,8,27,64,125

1³ =  1

2³ =  8

3³ = 27

12. Perfect Numbers

Sum of all the factors of a number except the number, is the number itself is called perfect number.

Eg.1)        6 – Factor of 6  = 1,2,3,6

                               6 =   1 x6

                               6 =   3 x2

Sum of factors except 6   = 1+2+3 = 6 (The Number itself)

Eg:2)  Factors of 28  = 1,2,4,7,14,28

                              28   = 1 x 28

                              28   = 14 x 2

                              28   = 7 x  4

Sum of factors except 28   = 1+2+4+7+14 = 28

13. Armstrong Numbers

            153,370,371,407 are Armstrong Numbers.

14. Ramanujan Number

Ramanujan Number       —–  1729

1729       —-  It is the smallest number expressible as the sum of two (positive)  cubes in two different ways.

          1729    = 103  + 93  = 1000 + 729  = 1729

          1729     = 123   + 13  = 1728 + 1      = 1729

The second number which is in the form of the Ramanujan number is 4104.

           4104      = 153  + 93   =  3375 + 729   = 4104

           4104       = 163  + 23  = 4096 + 8        = 4104

15. Numbers with Roman Representation

Eg:  15 =     10 + 5           =XV

       64  =   50 + 10 + 4     = L X IV

       167 =    100 + 60 + 7  = C LX VII

        99  =   90 + 9            = XC IX

       33  =   30 + 3           = XXX III

Sum of ‘n’ Numbers

• Sum of first ‘n’ natural numbers   =   n(n+1)/2

• Sum of first ‘n’ odd numbers   = n²

ie;

• Sum of first ‘n’ even numbers   = n (n+1)

   ie;

• Sum of  squares of first ‘n’ natural  numbers = 1/6n (n + 1)(2n +1)

ie ,

• Sum of cubes of first ‘n’ natural numbers  = (n(n+1)/2)²

    ie,  1³  + 2³ + 3³ +…………….=     (n(n+1)/2)²        =   1/4(n(n+1)/2)

                                2.Divisibilty

• Divisibility by 2

            A number is divisible by 2 , if it’s unit digit is any of  0,2,4,6 and 8.

            Eg;  362, 200, 124

• Divisibility by 3

           A number is divisible by 3, it the sum of it’s digit is divisible by 3.

           Eg; 321 = 3+2+1     =6 (divisible by3)

          So 321 is divisible by 3.

          Eg;   27356312 =  2+7+3+5+6+3+1+2   = 29

          29 ( not divisible by 3)

          So 27356312   is not divisible by 3.

• Divisiblity by 4

          A number is divisible by 4 , if the number formed by the last two digits is

          divisible by  4 or the last two digits are zeros.

           Eg;   600 :   Since last two digits are zeros

                       Therefore, 600 is divisible by 4.

           Eg;  4528 :   Since 28 is divisible by 4.

                          Therefore, 4528 is divisible by 4.

• Divisibility by 5

           A number  which is divisible by 5, if its unit’s digit ie either 5 or zero.

           Eg;  3775, 4050

• Divisibility by 6

           A number is divisible by 6, if it is  divisible by both 2 and 3.

           Eg;   612 : Divisible by both 2 and 3.                      612 ÷ 2 = 306

                       So divisible by 6.                                             612 ÷ 3 = 204

          Eg;   328 : Divisible by 2 but not by 3.

          So not divisible by 6                                                    328 ÷ 2 = 164

                                                                                                 328  ÷ 3 = 109.333

• Divisible by 8 

           A   number  is divisible by 8, if the  last 3 digits is divisible by 8 or

           the last 3 digits are zeros.

           Eg;   34000   : Last 3 digits are zeros                        34000 ÷ 8 = 4250

                              So divisible by 8.

            62128 : Last 3 digits =128

                              So   62128 is divisible by 8.

• Divisibility by 9                

            A number is divisible by 9, if the sum of its digits is divisible by 9.

             Eg;   864 =  8 + 6 + 4

                        = 18  ( 18 ÷ 9 =  2 )

            2763  = 2 + 7 + 6 + 3

                        = 18 ( Divisible by 9)

             324     = 3 + 2 + 4  =9

• Divisibility by 10

            A number is divisible by 10 if its unit digit  is 0.

            Eg;  520 , 65320.

• Divisibility by 11

           A number is divisible by 11 if the difference of the sum of  its digits at

           odd places and the sum of its digits at even places is either 0 or a

           number  divisible by 11.

           Eg;  231 -Sum of digits at odd  places = 1 + 2

                     Sum of digits at even places =  3 (here only one digit).

               Difference  3-3 =0

• Divisibility by 12

          A number is divisible by 12 if it is divisible by both  3 & 4

          Eg;    10224 , 672                                           10224   3 =     3408

                                                                                    10224  ÷ 4 =    2556

• Divisibilty by 14

           A number is divisible by 14, if it is divisible by both  2 and 7.

           Eg:   39200,   1988                                   39200 ÷ 2 = 19600

              39200 ÷ 7 =     5600

• Divisibilty by 16

           A number is divisible by 16,  if the last 4 digits are divisible by 16 or it is

           either  ‘0000’ .

           Eg;   10000,  60 5024 5024 ÷  16  = 314

• Divisibilty by 18

           A number is divisible by 18 if it is divisible by both 2 and 9.