Division on Number

Division Algorithm:

By Euclid's Division Algorithm a=bq+r

where

a=Dividend b=Divisor q=quotient r=remainder

b is not equal to zero

& 0<r<b

In general, we have :

Dividend = (Divisor X Quotient ) +Remainder .

Example: -

1) Suppose we divide 123 by 16. Then we have

16) 123 ( 7

-112

-------

Dividend = 123

Divisor = 16

Quotient = 7

Remainder = 11

Clearly, 123 = 16 X 7 +11

2) On dividing 16890 by a certain number, the quotient is 123 and the remainder is 39. Find the divisor.

Sol. Divisor = (Dividend - Remainder)

---------------------------------------

Quotient

= 16890 - 39

-----------------

123 = 137

3)Find the number nearest to 5028, which is exactly divisible by 64. Sol. 64) 5028 (78

- 448

-----------

548

- 512

-----------

Remainder 36, which is greater than half of the divisor. Therefore

Required number = 5028+(64 -36) = 5056 .

4) Find the number which is nearest to 6444 and exactly divisible by 42.

Sol. 42 ) 6444 ( 153

- 42 ------------

224

- 210

-----------

144

- 126

----------

Remainder 18, which is less than half of the divisor. Therefore

Required number = ( 6444 - 18 )

= 6426

5) What least number must be added to 5204 to get a number exactly divisible by 180 ?

Sol. 180)5204 ( 28

-360

--------

1604

- 1440

---------

164

Remainder 164 , which is nearest to divisor.

Therefore

Required number = (180 - 164 )

= 16

6)what least number must be subtracted from 6500 to get a number exactly divisible by 135 ?

Sol. 135 )6500 (48

- 540

-------

1100

- 1080

-------

Remainder is 20

Therefore the number to be subtracted is 20.

Tests of Divisibility

1) Divisibility by 2:-

If unit digit of given number is any one of 0,2,4,6 & 8.

Example -

84954 is divisible by 2 ,while 54321 is not.

2) Divisibility by 3 :-

If the sum of digits of given number is divisible by 3.

Example:-

592482 is divisible by 3, sum of its digits 5+9+2+4+8+2 =30 which is divisible by 3.

3)Divisibility by 4 :-

A number is divisible by 4, if the number formed by the last two digit is divisible by 4.

Example :-

892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4.

4)Divisibility by 5 :-

A number is divisible by 5, if its unit's digit is either 0 or 5 .

Example :-

20820 and 50345 are divisible by 5 , while 30934 and 40946 are not.

5) Divisibility by 6 :-

A number is divisible by

6 , if it is divisible by both 2 and 3 .

Example :-

The number 35256 is divisible by 2 . Sum of its digits = ( 3+5+2+5+6 ) = 21 , which is divisible by 3 . thus , 35256 is divisible by 2 as well as 3 .Hence , 35256 is divisible by 6 .

6) Divisibility by 8 :-

A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8 .

Example :-

953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8 .

While , 529418 is not divisible by 8 , since the number formed by last three digits is 418 , which is not divisible by 8.

7) Divisibility by 9 :-

A number is divisible by 9 , if the sum of its digits is divisible by 9.

Example :-

60732 is divisible by 9, since sum of digits = ( 6+0+7+3+2 ) = 18 , which is divisible by 9.

But , 68956 is not divisible by 9 , since sum of digits = ( 6+8+9+5+6 ) = 34 , which is not divisible by 9 .

8) Divisibility by 10 :-

A number is divisible by 10 , if it ends with 0 .

Example :-

96410 , 10480 are divisible by 10, while 96375 is not .

9) Divisibility by 11 :-

A number is divisible by 11 , if the difference of the sum of its digits at odd places and the sum of its digits at even places , is either 0 or a number divisible by 11.

Example :-

The number 4832718 is divisible by 11 , since : ( sum of digits at odd places ) - ( sum of digits at even places ) = ( 8+7+3+4 ) - ( 1+2+8 ) = 11, which is divisible by 11

10) Divisibility by 12 :-

A number is divisible by 12 , if it is divisible by both 4 and 3 .

Example :-

Consider the number 34632 .

(a) The number formed by last two digits is 32 , which is divisible by 4.

(b) Sum of digits = ( 3+4+6+3+2 ) =18 which is divisible by 3 .

11) Divisibility by 14 :-

A number is divisible by 14 , if it is divisible by 2 as well as 7.

Example :-

42 ,56

12) Divisibility by 15 :-

A number is divisible by 15 , if it is divisible by both 3 and 5.

Example :-

30 ,45 ,120

13) Divisibility by 16 :-

A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16.

Example :-

7957536 is divisible by 16 , since the number formed by the last four digits is 7536 , which is divisible by 16 .

14) Divisibility by 24 :-

A given number is divisible by both 3 and 8.

15) Divisibility by 40 :-

A given number is divisible by 40, if it is divisible by both 5 and 8.

16) Divisibility by 80 :-

A given number is divisible by 80 ,if it is divisible by both 5 and 16 .

Note :-

If a number is divisible by p as well as q , where p and q are co- primes ,then the given number is divisible by p and q .

If p and q are not co- primes , then the number need not be divisible by p and q , even when it is divisible by both p and q .

Example :-

36 is divisible by both 4 and 6 ,

but it is not divisible by ( 4×6 ) = 24, since 4 and 6 are not co- primes .

-------- Exercise -----

1) Which of the following numbers is divisible by 3 ?

a)541326 b)5967013

2) Which of the following numbers is divisible by 4 ?

a)67920594 b)618703572 Answer 1) a 2) b