PROPERTIES OF DIVISION

Property 1 :

When a number 'x' is divided by another number 'y', the number 'x' is divided into 'y' number of equal parts. 

If 'y' divides 'x' without any remainder, then 'x' is evenly divisible by 'y'. 

Examples : 

(i)  When 21 is divided by 3, 21 is divided into three equal parts and the value of each part is 7.  

(ii)  When we divide 35 by 5, we get

7 + 7 + 7 + 7 + 7

(iii)  When 35 is divided by 5, 35 is divided into 5 equal parts and the value of each part is 7. And also, there is nothing left over in 35.

So, 35 is evenly divisible by 5.

(iv)  When 35 is divided by 4, we get

8 + 8 + 8 + 8 + 3

There is remainder 5, when 35 is divided by 3.

So, 35 is not evenly divisible by 4.

Property 2 : 

When a number is divided by another number, the division algorithm is, the sum of product of quotient & divisor and the remainder is equal to dividend.

More clearly, 

Dividend  =  Quotient x Divisor + Remainder

The number which we divide is called the dividend.

The number by which we divide is called the divisor.

The result obtained is called the quotient.

The number left over is called the remainder.

Property 3 :

When a number is divided 1, the quotient is the number itself. 

Example : 

7/1  =  7

Property 4 :

When a number is divided by itself, the quotient is 1.

Example : 

5/5  =  1

Property 5 :

When we divide any non-zero number by zero, the quotient is undefined. So, dividing any non-zero number by zero is meaningless.  

Example : 

3/0  =  Undefined

Property 6 :

When zero is divided by any non-zero number, the quotient is zero. 

Example : 

0/5  =  0  

Property 7 : 

When a number is divided by another number which is a multiple of 10 like 10, 100, 1000 etc., the decimal point in the number has to be moved to the left. 

Examples : 

123/10  =  12.3

2.36/100  =  0.0236

5658.36/1000  =  5.65836

Property 8 : 

Positive number / Positive number  =  Positive number

Negative number / Negative number  =  Positive number

Negative number / Positive number  =  Negative number

Positive number / Negative number  =  Negative number

Note : 

(i)  Division is not commutative.

Example : 

15 ÷ 5  =  15/5  =  3

5 ÷ 15  =  5/15  =  1/3

Therefore, 

15 ÷ 5  ≠  5 ÷ 15

(ii)  Division is not associative property. 

Example : 

12 ÷ (6 ÷ 2)  =  12 ÷ 3  =  4

(12 ÷ 6) ÷ 2  =  2 ÷ 2  =  1

Therefore, 

12 ÷ (6 ÷ 2)  ≠  (12 ÷ 6) ÷ 2

Problem 1 :

Division is the repeated_____________

i) addition        (ii) subtraction

(iii) multiplication     (iv) none of these

Solution :

Division is the repeated subtraction.

Problem 2 :

We cannot divide any number by ______

i) 0       (ii) 1     (iii) itself      (iv) none of these

Solution :

• While dividing any number by 0, we will get undefined.
• While dividing any number by 1, we will get the same number.
• Dividing a number by itself, we will get the 1.

So, none of these is the answer.

Problem 3 :

36 ÷ 6 =_____

Solution :

The number of 6's in 36 is 6. So, 36 ÷ 6 = 6

Problem 4 :

In the division 50 ÷ 5 dividend is _____.

Solution :

The number of 5's in 50 is 10. So, 50 ÷ 5 = 10

Problem 5 :

13 ÷____= 13

Solution :

Dividing a number by itself, we will get 1.

Dividing a number by 1, we will get the same number.

Since we get 13 as the result, the given number 13 should be divided by 1.

13 ÷ 1 = 13

Problem 6 :

If 8 X 9 = 72 then 72 ÷ ___= 9

Solution :

8 X 9 = 72

Multiplying these two numbers, we get 72. By dividing the result by any one of the number, we will get the other number.

So, 72 ÷ 8 = 9

Problem 6 :

9 X 100 =____

Solution :

Multiplying 9 and 100, we will get 900.

Problem 7 :

0 ÷ 81 =___

Solution :

Dividing 0 by any number, we get 0.

Problem 8 :

There are 56 toffees in a packet. If the toffees are distributed among 7 children then how much each one will get?

Solution :

Total number of toffees in a packet = 56

Number of children who were getting = 7

Amount of toffees each may get = 56/7

= 8

So, each children will receive 8 toffees. 

Problem 9 :

If a person distribute 650 bags among 13 people then how much each one will get?

Solution :

Total number of bags = 650

Number of people = 13

Amount each will receive = 650/13

= 50

So, each will receive 50 bags.

Problem 10 :

A pizza had 8 slices and Ria ate half of the pizza during lunch time. How many slices of pizza did she eat?

Solution :

Number of slices = 8

Eating half of the slices = 8 / 2 

= 4 slices

So, 4 slices she has eat.

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