Exercise 5.2- Playing with Numbers RD Sharma Solutions | Mathematics (Maths) Class 8 PDF Download

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Q u e s t i o n : 1 1
If x denotes the digit at hundreds place of the number 
¯
67x19
such that the number is divisible by 11. Find all possible values of x.
S o l u t i o n :
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 multiple of 11. Sum of digits at odd places -Sum
Q u e s t i o n : 1 2
Find the remainder when 981547 is divided by 5. Do this without doing actual division.
S o l u t i o n :
If a natural number is divided by 5, it has the same remainder when its unit digit is divided by 5. Here, the unit digit of 981547 is 7. When 7 is divided by 5, remainder is 2. Therefore, remainder
Q u e s t i o n : 1 3
Find the remainder when 51439786 is divided by 3. Do this without performing actual division.
S o l u t i o n :
Sum of the digits of the number 51439786 = 5 + 1 + 4 + 3 + 9 + 7 + 8 + 6 = 43The remainder of 51439786, when divided by 3, is the same as the remainder when the sum of the digits
Q u e s t i o n : 1 4
Find the remainder, without performing actual division, when 798 is divided by 11.
S o l u t i o n :
798 = A multiple of 11 + (Sum of its digits at odd places - Sum of its digits at even places)798 = A multiple of 11 + (7 + 8 - 9)798 = A multiple of 11 + (15 - 9)798 = A multiple of 11 +
Q u e s t i o n : 1 5
Without performing actual division, find the remainder when 928174653 is divided by 11.
S o l u t i o n :
928174653 = A multiple of 11 + (Sum of its digits at odd places -Sum of its digits at even places)928174653 = A multiple of 11 + {(9 + 8 + 7 + 6 + 3) - (2 + 1 + 4 + 5)}928174653 = 
Q u e s t i o n : 1 6
Given an example of a number which is divisible by
i
2 but not by 4.
ii
3 but not by 6.
iii
4 but not by 8.
iv
both 4 and 8 but not by 32.
S o l u t i o n :
i
10
Every number with the structure (4n + 2) is an example of a number that is divisible by 2 but not by 4.
ii
15
Every number with the structure (6n + 3) is an example of a number that is divisible by 3 but not by 6.
iii
28
Every number with the structure (8n + 4) is an example of a number that is divisible by 4 but not by 8.
iv
8
Every number with the  structure (32n + 8), (32n + 16) or (32n + 24) is an example of a number that is divisible by 4 and 8 but not by 32.
Q u e s t i o n : 1 7
Which of the following statements are true?
i
If a number is divisible by 3, it must be divisible by 9.
ii
If a number is divisible by 9, it must be divisible by 3.
iii
If a number is divisible by 4, it must be divisible by 8.
iv
If a number is divisible by 8, it must be divisible by 4.
v
A number is divisible by 18, if it is divisible by both 3 and 6.
vi
If a number is divisible by both 9 and 10, it must be divisible by 90.
vii
If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.
viii
If a number divides three numbers exactly, it must divide their sum exactly.
ix
If two numbers are co-prime, at least one of them must be a prime number.
x
The sum of two consecutive odd numbers is always divisible by 4.
S o l u t i o n :
i
False
Every number with the structures (9n + 3) or (9n + 6) is divisible by 3 but not by 9. Example: 3, 6, 12 etc.
ii
True
iii
Page 3


                  
                                   
 
                                
     
             
                                    
                                 
                   
             
               
    
               
                                        
         
        
                                    
        
           
                                   
    
               
                                    
 
               
                                    
 
                     
                                        
Q u e s t i o n : 1 1
If x denotes the digit at hundreds place of the number 
¯
67x19
such that the number is divisible by 11. Find all possible values of x.
S o l u t i o n :
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 multiple of 11. Sum of digits at odd places -Sum
Q u e s t i o n : 1 2
Find the remainder when 981547 is divided by 5. Do this without doing actual division.
S o l u t i o n :
If a natural number is divided by 5, it has the same remainder when its unit digit is divided by 5. Here, the unit digit of 981547 is 7. When 7 is divided by 5, remainder is 2. Therefore, remainder
Q u e s t i o n : 1 3
Find the remainder when 51439786 is divided by 3. Do this without performing actual division.
S o l u t i o n :
Sum of the digits of the number 51439786 = 5 + 1 + 4 + 3 + 9 + 7 + 8 + 6 = 43The remainder of 51439786, when divided by 3, is the same as the remainder when the sum of the digits
Q u e s t i o n : 1 4
Find the remainder, without performing actual division, when 798 is divided by 11.
S o l u t i o n :
798 = A multiple of 11 + (Sum of its digits at odd places - Sum of its digits at even places)798 = A multiple of 11 + (7 + 8 - 9)798 = A multiple of 11 + (15 - 9)798 = A multiple of 11 +
Q u e s t i o n : 1 5
Without performing actual division, find the remainder when 928174653 is divided by 11.
S o l u t i o n :
928174653 = A multiple of 11 + (Sum of its digits at odd places -Sum of its digits at even places)928174653 = A multiple of 11 + {(9 + 8 + 7 + 6 + 3) - (2 + 1 + 4 + 5)}928174653 = 
Q u e s t i o n : 1 6
Given an example of a number which is divisible by
i
2 but not by 4.
ii
3 but not by 6.
iii
4 but not by 8.
iv
both 4 and 8 but not by 32.
S o l u t i o n :
i
10
Every number with the structure (4n + 2) is an example of a number that is divisible by 2 but not by 4.
ii
15
Every number with the structure (6n + 3) is an example of a number that is divisible by 3 but not by 6.
iii
28
Every number with the structure (8n + 4) is an example of a number that is divisible by 4 but not by 8.
iv
8
Every number with the  structure (32n + 8), (32n + 16) or (32n + 24) is an example of a number that is divisible by 4 and 8 but not by 32.
Q u e s t i o n : 1 7
Which of the following statements are true?
i
If a number is divisible by 3, it must be divisible by 9.
ii
If a number is divisible by 9, it must be divisible by 3.
iii
If a number is divisible by 4, it must be divisible by 8.
iv
If a number is divisible by 8, it must be divisible by 4.
v
A number is divisible by 18, if it is divisible by both 3 and 6.
vi
If a number is divisible by both 9 and 10, it must be divisible by 90.
vii
If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.
viii
If a number divides three numbers exactly, it must divide their sum exactly.
ix
If two numbers are co-prime, at least one of them must be a prime number.
x
The sum of two consecutive odd numbers is always divisible by 4.
S o l u t i o n :
i
False
Every number with the structures (9n + 3) or (9n + 6) is divisible by 3 but not by 9. Example: 3, 6, 12 etc.
ii
True
iii
False
Every number with the structure (8n + 4) is divisible by 4 but not by 8. Example: 4, 12, 20 etc.
iv
True
v
False
Example: 24 is divisible by both 3 and 6 but it is not divisible by 18.
vi
True
vii
False
Example: 5 divides 10, which is a sum of 3 and 7. However, it neither divides 3 nor 7.
viii
True
ix
False
Example: 4 and 9 are co-prime numbers but both are composite numbers too.
x
True