ML AGGARWAL CLASS 8 CHAPTER 5 Playing with Numbers Exercise 5.3

Exercise 5.3

Question 1

Sol :
(i) 87035

it is divisible by 5 


(ii) 75060

It is divisible by both 5 and 10 


(iii) 9685

It is divisible by 5 


iv. 10730

It is divisible by both 5 and 10 



Question 2

i. 67894

It is divisible by '2'


ii. 5673244

It is divisible by both 2 and 4


iii. 9685048

It is divisible by 2 , 4 and 8


iv. 75379

It neither divisible by 4 nor by 8 



Question 3

Sol :

(i) 45639

sum of digits  = 4 + 5 + 6 + 3 + 9 =27

27 is divisible by both 3 and 9

∴ 45639 is divisible by both 3 or 9 


(ii) 301248

Sum of digits = 3 + 0 + 1 + 2 + 4 + 8 = 18

18 is divisible by 9

∴ . 301248 is divisible by both 3 or 9 


(iii) 567081

Sum of digits =5+6+1+0+8+1=27

27 is divisible by 9

∴ 567081 is divisible by either 3 or by 9 


(iv)  345903

sum of digits =3+4+5+9+0+3=24

24 is divisible by only'3'

∴ 345903  is divisible by '3'only.


(v) 345046

sum of digits =3+4+5+0+4+6=22

22 is neither divisible by 3 nor by 9

∴ 345046 is neither divisible by 3 nor by 9 



Question 4

Sol :

(i) 10835

Sum of odd displace digits =5+8+1=14

Sum of even place digits =3+0=3

Difference =14-3=11

∴ 10835 is divisible by 11


(ii) 380237

Sum of odd place digits =1+2+8=17

Sum of even place digits =3+0+3=6

Difference =17-6=11

∴ 380237 is divisible by 11


(iii) 504670

Sum of odd place digits =0+6+0=6

Sum of even place digits =7+4+5=16

Difference = 16 - 6 = 10 

∴ 504670 is not divisible by 11 


(iv) 28248

Sum of odd place digits =8+2+2=12

Sum of even place digits =4+8=12

difference =12-12=0

 ∴ 28248 is divisible by 11 


Question 5

Sol :

(i)15414

15414 is divisible by '2' because units place contain '4'

1+5+4+1+4=15 divisible by 3

  ∴ 15414 in divisible by 3

  ∴ 15414 is divisible by 6


(ii) 213888

213888 is divisible by 2

Sum of digits =2+1+3+8+8+8=30 divisible by'3

  ∴ 213888 is  also divisible by 3

  ∴ 213888 is divisible by ' 6 '


(iii) 469876

469876 is divisible by 2

Sum of digits =4+6+9+8+7+6=40 not divisible by '3'

  ∴ 469876 in not divisible by 3

  ∴ 469876 in also not divisible by 6 '.



Question 6

Sol :
(i) 4618894875

Sum of digits of alternative blocks

875+618=1493  and 894+4=898

Difference $=595$ which is divisible by 7

   ∴ 4618894875 is divisible by 7 


(ii) 3794856

Sum of digits of alternative blocks

856+3=859 and 794

difference = 65 which is not divisible by 7 


(iii) 39823

sum of digits of alternative blocks

823 and 39

Difference 823-39=784 which is divisible by 7

 ∴ 3794856 is not divisible by 7 


(iii) 39823

sum of digits of alternative blocks

823 and 39

Difference 823-39=784 which is divisible by 7

 ∴  39823 is divisible by ' 7 '.




Question 7

Sol :

(i) Given number = 34x 

if 34x is multiple of 3 then (3+ 4 + x ) should be multiple of '3'

3+ 4+ x = 9+x

so to make sum of digits multiple of 3 
x should be 0 , 3 , 6 , 9 


(ii) 74 x 5284 is multiple of '3 'so

7+4+x+5+2+844=30+x

So to make sum of digits multiple of 3
x should be 0,3,6,9 .



Question 8

Sol :

Given number 43Z3 

If 42 z 3 in multiple of  then (4+2+z+3)
Should be multiple of 9

4+2+z+3=9+z

To make sum of digits multiple of a
should be 0,9 .


   

Question 9

Sol :


(i)  49*2207

If a number is divisible by 9 then its sum of digits 
should be divisible by 9 

4 + 9 + * + 2 + 2+ 0 + 7 = 24+*

The near by multiple of 9 is 27 so in place of the digit must be 27- 24 = 3


(ii) 5938*623

"If a number is divisible by 9 then it's sum of digits
Should be divisible by 9

5 + 9 + 5 + 8 + * + 6 + 2 + 3= 36 + *

36 is multiple of 9 so * should be either 0 or 9 


Question 10

Sol :

(i) 97*542

As unit's place is 2 , number in divisible by 2
irrespective of digit in * 'Place.

if a number divisible by 3 then its sum of digits should be divisible by 3

9+7-1 x+5+4+2=27+x

∴ 27 in a multiple of 3 so in place of '*' Should be

either 0,3,6, or 9

In place of * 0 , 3 , 6 , 9 number makes given 
number divisible by 6 (As it is divisible by both 2 and 3)


(ii)  709*94

As units digit in 4 given number divisible by
2  irrespective of number  in place of *

To make given number divisible by ,3 the sum of digits should be divisible by 3 

7+0+9+*+9+4=29+*

To make divisible $29, *$ should be either 1,4,7 

$\therefore$ In place ot *, 1,4 or 7 digits make the given number divisible by 6. (As it is divisible by 2 \  and 3


Question 11

Sol :
(i) given number $64 * 2456$

Sum of digits in odd places =6+4+*+6=16+*
Sum of digits in even places =5+2+4=11

Difference $=16+*-11=5+*$

* Should  Because to make given number
divisible by 11 , the difference in sum of digits in even
and odd place is either 0 or multiple of 11 

∴ should be 6


(ii) 86*6194

sum of digits in even places = 9 + 6+ 6= 21

sum of digits in odd places = 4+ 1 + * + 8 = 13+ *

so the difference should be either 0 or multiple 
of 11 to be divisible by 11

8 - * =0

* = 8

∴ to make 86*6194 divisible by 11 , in place of * digits must be place 

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