# 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  ≠  ÷ 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

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