NUMBER SYSTEM

 Number System is divided into 4 parts:

1. Classification

2. Divisibility Test

3. Division and Remainder Rules

4. Sum Rules 

In this e-book we will look into some tricks that will make the solving of the questions involving 

number system easy.

TRICKS TO REMEMBER FOR NUMBER SYSTEM: 

1. Classification

Types Description

Natural Numbers all counting numbers ( 1,2,3,4,5….∞)

Whole Numbers natural number + zero( 0,1,2,3,4,5…∞)

Integers

All whole numbers including Negative number + Positive 

number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞)

Even & Odd Numbers

All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and 

which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞)

Prime Numbers

It can be positive or negative except 1, if the number is not divisible by 

any number except the number 

itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞)

Composite Numbers Natural numbers which are not prime.

Co-Prime Two natural number a and b are said to be co-prime if their HCF is 1.

2. Divisibility

Numbers IF A Number

Divisible by 2 End with 0,2,4,6,8 are divisible by 2

Divisible by 3 Sum of its digits is divisible by 3

Divisible by 4 Last two digit divisible by 4

Divisible by 5 Ends with 0 or 5

Divisible by 6 Divides by Both 2 & 3

Divisible by 8 Last 3 digit divide by 8

Divisible by 10 End with 0

Divisible by 11 [Sum of its digit in odd places-Sum of its 

digits in even places]= 0 or multiple of 11

Divisible by 12 [The number must be divisible by 3 and 

4]

Divisible by 13

[Multiply last digit with 4 and add it to 

remaining number in given number, 

result must be divisible by 13]

Divisible by 14

[The number must be divisible by 2 and 

7. Because 2 and 7 are prime factors of 

14.]

Divisible by 15

[The number should be divisible by 3 and 

5. Because 3 and 5 are prime factors of 

15.]

Divisible by 16 [The number formed by last four digits in 

given number must be divisible by 16.]

Divisible by 17

[Multiply last digit with 5 and subtract it 

from remaining number in given 

number, result must be divisible by 17]

Divisible by 18 [The number should be divisible by 2 and 

9]

Divisible by 19

[Multiply last digit with 2 and add it to 

remaining number in given number, 

result must be divisible by 19]

Divisible by 20 [The number formed by last two digits in 

given number must be divisible by 20.]

3. Division & Remainder Rules

A very basic formula for division rules is:

dividend = ( divisor✘quotient ) + remainder

or

divisor= [(dividend)-(remainder] / quotient

This could be mathematically written in another way:

 x = kq + r where (x = dividend, k = divisor, q = quotient, r = remainder).

4. Sum Rules:

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

• Sum of square of first n natural numbers= n(n+1)(2n+1)/6

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

• Sum of first n odd numbers= n^2

• Sum of first n even numbers= n(n+1)

Now we will look into basic mathematical formulas that help in solving the Number System

questions very frequently.

1. (a + b)(a - b) = (a2

 - b2

)

2. (a + b)2 = (a2 + b2 + 2ab)

3. (a - b)2 = (a2 + b2 - 2ab)

4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

5. (a3 + b3

) = (a + b)(a2

 - ab + b2

)

6. (a3 - b3) = (a - b)(a2 + ab + b2)

7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)

8. (a+b)³=a³+3a²b+3ab²+b³

9. (a-b)³=a³-3a²b+3ab²-b³