CHAPTER – 2

NUMBERS


 

NUMBERS is the most important topic for Various Competitive Exams. In this chapter, we have put together some model problems that have been appearing in different exams.

For understanding certain mathematical concepts, it is necessary to learn certain categorizations of numbers. The categories of numbers are

Natural Numbers: All counting numbers are Natural numbers that start from 1.

                               Example: 12345…….

 

Even Numbers and odd Numbers: The numbers which Are exactly divisible by 2 are even numbers and the number which leaves one as a remainder divided by 2 is called ODD NUMBERS.

Eg: 2,4,6, 8 ……. are Even Numbers.

Eg: 1,3,5,7,9 ……. are ODD numbers

Prime Numbers: The numbers which are only divisible by 1 and themselves are prime numbers...

Eg : 2,3,5,7,11,13…..

composite numbers: Numbers that are divisible by other numbers along with one and themselves are composite...

Eg: 4,6,8,9,10……

Whole Numbers: The natural numbers along with 0 is called as Whole Number

Eg: 0,1,2,3,4,5,…..

Whole numbers Are denoted by Z.

Integers:  Any whole numbers which are positive, negative, or Zero are nothing but Integers …

Eg : { ……..-4,-3,-2,-1 , 0 ,+1,+2,+3,+4…….}

Rational Numbers:  Any numbers which can be expressed as a/b are integers and b ≠ 0, is a Rational Numbers. In other words; rational numbers include integers.

Eg: 4/5, 4, -6, 23/7, etc...

Irrational Numbers: A number that can not be expressed in a/b form or any fraction are called irrational numbers

Eg : 3,6,16

Real numbers: The group of rational and irrational numbers is called a set of real numbers...

Factors :



A number which is expressed as a product of Two or more numbers, then the numbers which are multiplied are called factors ,

Eg: Find out Factors of 12

 2   ×  6

3    ×  4 are the Factors of 12.

Co – prime Numbers:

If two numbers are such that there is no common factor between them, then they are said to be relatively co-prime with each other.

Eg : ( 15,23) , (12,13), (4,15) etc

Divisibility Rules:

A number to be divisible by 2 has to have its  last digit even numbers (136,144)

A number is divisible by 3 if the sum of its digits are multiples of 3

Eg: 126= 1+2+6 = 9 which is divisible by 3

A Number is divisible by 4 if its last two digits are divisible by 4.

Eg: 1236, 23480

A number which is divisible by 5 if its last digit is 5 or 0

Eg: 3965, 3565

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

Eg: 18, 42 , 36, 48 etc

A number which is divisible by 8 if its last  3 digits are divisible by 8 only

Eg: 3816, 14328,18864 etc

A number is divisible by 9, if sum of its digits equals to 9 or is a multiple of 9

Eg : 18 = 1+8 = 9

        729 = 7+2+9 = 18

A  number which is divisible by 11, if the sum of the alternate digits is the same or multiple of 11

Eg: 132 sum of alternate digits is  1+2 = 3 and 3 which are equal, hence it is a multiple of 11.