PROPERTIES OF DIVISION OF INTEGERS

In Math, the whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z  =  {... - 2, - 1,0,1,2, ...}, is the set of all integers

Let us look at the properties of division of integers. 

Property 1 :

When an integer 'x' is divided by another integer 'y', the integer '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 integer is divided by another integer, 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 integer which we divide is called the dividend.

The integer by which we divide is called the divisor.

The result obtained is called the quotient.

The integer left over is called the remainder.

Property 3 :

When an integer is divided 1, the quotient is the number itself. 

Example : 

7/1  =  7

Property 4 :

When an integer is divided by itself, the quotient is 1.

Example : 

5/5  =  1

Property 5 :

When we divide any positive or negative integer by zero, the quotient is undefined. So, dividing any positive or negative integer by zero is meaningless.  

Example : 

3/0  =  Undefined

Property 6 :

When zero is divided by any positive or negative integer, the quotient is zero. 

Example : 

0/5  =  0  

Property 7 : 

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

Examples : 

123/10  =  12.3

123/100  =  1.23

123/1000  =  0.123

123/10000  =  0.0123

Property 8 : 

Positive integer / Positive integer  =  Positive value

Negative integer / Negative integer  =  Positive value

Negative integer / Positive integer  =  Negative integer

Positive integer / Negative integer  =  Negative value

Closure Property of Division of Integers

Observe the following examples :

(i)  15 ÷ 5  =  15/5  =  3

(ii)  (-3)  ÷  9  =  -3/9  =  -1/3  

(i)  7 ÷ 4  =  7/4  =  1.75

(ii)  1  ÷  2  =  1/2  =  0.5  

From the above examples we observe that integers are not closed under division.

Commutative Property of Division of Integers

Observe the following examples :

15 ÷ 5  =  15/5  =  3

5 ÷ 15  =  5/15  =  1/3

Therefore, 

15 ÷ 5  ≠  5 ÷ 15

From the above example, we observe that integers are not commutative under division.

Associative Property of Division of Integers

Observe the following examples :

12 ÷ (6 ÷ 2)  =  12 ÷ 3  =  4

(12 ÷ 6) ÷ 2  =  2 ÷ 2  =  1

Therefore,

12 ÷ (6 ÷ 2)  ≠  (12 ÷ 6) ÷ 2

From the above example, we observe that integers are not associative under division.

Zero Division Property

Division of any non-zero number by zero is meaningless.

Because division by zero is not defined.

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