Well-designed AP 6th Class Maths Solutions Chapter 3 Playing with Numbers Exercise 3.3 offers step-by-step explanations to help students understand problem-solving strategies.
Playing with Numbers Class 6 Exercise 3.3 Solutions – 6th Class Maths 3.3 Exercise Solutions
Question 1.
Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11 (say, yes or no) :
Solution:
Question 2.
Using divisibility tests, determine which of the following numbers are divisible by 4; by 8 :
a) 572
Solution:
Given number = 572
i) Divisibility by 4
The number formed by the last two digits of the given number = 72
Remainder = 0
Hence, 572 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 572
Remainder = 4
Hence, 572 is not divisible by 8.
b) 726352
Solution:
Given number 726352
i) Divisibility by 4
The number formed by the last two digits of the given number = 52
Remainder = 0
Hence, 726352 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 352
Remainder = 0
Hence, 726352 is divisible by 8.
c) 5500
Solution:
Given number 5500
i) Divisibility by 4
The number formed by the last two digits of the given number = 00
Hence, 5500 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 500
Remainder = 4
Hence, 5500 is not divisible by 8.
d) 6000
Solution:
Given number 6000
i) Divisibility by 4
Here, the last two digits of the given number are 00.
Hence, 6000 is divisible by 4.
ii) Divisibility by 8
Here, the last three digits of the given number are 000.
Hence, 6000 is divisible by 8.
e) 12159
Solution:
Given number 12159
i) Divisiblity by 4
The number formed by the last two digits of the given number = 59
Remainder = 3
Hence, 12159 is not divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 159
Remainder = 7
Hence, 12159 is not divisible by 8.
f) 14560
Solution:
Given number 14560
i) Divisibility by 4
The number formed by the last two digits of the given number = 60
Remainder = 0
Hence, 14560 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 560
Remainder = 0
Hence, 14560 is divisible by 8.
g) 21084
Solution:
Given number 21084
i) Divisiblity by 4
The number formed by the last two digits of the given number = 84
Remainder = 0
Hence, 21084 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 084
Remainder = 4
Hence, 21084 is not divisible by 8.
h) 31795072
Solution:
Given number 31795072
i) Divisibility by 4
The number formed by the last two digits of the given number = 72
Remainder = 0
Hence, 31795072 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 072
Remainder = 0
Hence, 31795072 is divisible by 8.
i) 1700
Solution:
Given number 1700
i) Divisibility by 4
Here, the last digits of the given number are 00.
Hence, 1700 is divisible by 4.
ii) Divisibility by 8
The number formed by the last three digits of the given number = 700
Hence, 1700 is not divisible by 8.
j) 2150
Solution:
Given number 2150
i) Divisibility by 4
The number formed by the last two digits of the given number = 50
Remainder = 2
Hence, 2150 is not divisible by 4.
ii) Divisiblity by 8
The number formed by the last three digits of the given number = 150
Remainder = 6
Hence, 2150 is not divisible by 8.
Question 3.
Using divisibility tests, determine which of the following numbers are divisible by 6 :
a) 297144
Solution:
We know that, if a number is divisible by 2 and 3 both, then it is divisible by 6 also.
Given number = 297144
The digit at one’s place of the given number = 4
So, it is divisible by 2.
The sum of all digits of the given number = 2+9+7+1+4+4=27 which is divisible by 3.
Hence, 297144 is divisible by 6.
b) 1258
Solution:
Given number = 1258
The digit at one’s place of the given number = 8
So, it is divisible by 2.
The sum of all digits of the given number = 1+2+5+8=16 which is not divisible by 3.
Hence, 1258 is not divisible by 6.
c) 4335
Solution:
Given number = 4335
The digit at one’s place of the given number = 5
So, it is not divisible by 2.
Hence, 4335 is not divisible by 6.
d) 61233
Solution:
Given number = 61233
The digit at one’s place of the given number is 3
So, it is not divisible by 2.
Hence, 61233 is not divisible by 6.
e) 901352
Solution:
Given number =901352
The digit at one’s place of the given number is 2
So, it is divisible by 2.
The sum of the all digits of the given number 901352 = 9 + 0 + 1 + 3 + 5 + 2 = 20
which is not divisible by 3.
Hence, it is not divisible by 6.
f) 438750
Solution:
Given number = 438750
The digit at one’s place of the given number is 0
So, it is divisible by 2.
The sum of the all digits of the given number 438750 = 4 + 3 + 8 + 7 + 5 + 0 = 27
which is divisible by 3.
Hence, the given number is divisible by 6.
g) 1790184
Solution:
Given number = 1790184
The digit at one’s place of the given number is 4
So, it is divisible by 2.
The sum of the all digits of the given number 1790184 = 1 + 7 + 9 + 0 + 1 + 8 + 4 = 30
which is divisible by 3.
Hence, the given number is divisible by 6.
h) 12583
Solution:
Given number = 12583
The digit at one’s place of the given number is 3
So, it is not divisible by 2.
Hence, the given number is not divisible by 6.
i) 639210
Solution:
Given number = 639210
The digit at one’s place of the given number is 0
So, it is divisible by 2.
The sum of the all digits of the given number 639210 = 6+3+9+2+1+0 = 21
which is divisible by 3.
Hence, the given number is divisible by 6.
j) 17852
Solution:
Given number = 17852
The digit at one’s place of the given number is 2
So, it is divisible by 2.
The sum of the all digits of the given number 17852 = 1 + 7 + 8 + 5 + 2 = 23
which is not divisible by 3.
Hence, the given number is not divisible by 6.
Question 4.
Using divisibility tests, determine which of the following numbers are divisible by 11 :
a) 5445
Solution:
We know that a number is divisible by 11, If the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number is either 0 or divisible by 11.
a) Given number = 
Sum of the digits at odd places = 5 + 4 = 9
Sum of the digits at even places = 4 + 5 = 9
Difference = 9 – 9 = 0
Hence, the given number is divisible by 11.
b) 10824
Solution:
Given number = 
Sum of the digits at odd places = 4 + 8 + 1 = 13
Sum of the digits at even places = 2 + 0 = 2
Difference = 13 – 2 = 11, which is divisible by 11.
Hence, the given number is divisible by 11.
c) 7138965
Solution:
Given number = 
Sum of the digits at odd places = 5 + 9 + 3 + 7 = 24
Sum of the digits at even places = 6 + 8 + 1 = 15
Difference = 24 – 15 = 9, which is not divisible by 11.
Hence, the given number is not divisible by 11.
d) 70169308
Solution:
Given number = 
Sum of the digits at odd places = 8 + 3 + 6 + 0 = 17
Sum of the digits at even places = 0 + 9 + 1 + 7 = 17
Difference = 17 – 17 = 0
Hence, the given number is divisible by 11.
e) 10000001
Solution:
Given number = 
Sum of the digits at odd places = 1 + 0 + 0 + 0 = 1
Sum of the digits at even places = 0 + 0 + 0 + 1 = 1
Difference = 1 – 1 = 0
Hence, the given number is divisible by 11.
f) 901153
Solution:
Given number = 
Sum of the digits at odd places = 3 + 1 + 0 = 4
Sum of the digits at even places = 5 + 1 + 9 = 15
Difference = 15 – 4 = 11
Hence the given number is divisible by 11.
Question 5.
Write the smallest digit and the greatest digit in the blank space of each of the following numbers so that the number formed is divisible by 3;
a) _6724
Solution:
We know that number is divisible by 3, if the sum of all the digits of the number is also divisible by 3.
a) _6724.
Sum of the given digits = 6 + 7 + 2 + 4 = 19
if the smallest digit to be placed in blank space = 2
Then the sum = 19 + 2 = 21 which is divisible by 3.
Next, If the greatest digit to be placed in blank space = 8
Then the sum = 19 + 8 = 27 which is divisible by 3.
Hence, the required smallest digit = 2 and greatest digit = 8
b) 4765_2
Solution:
4765 _ 2
Sum of the given digits = 4 + 7 + 6 + 5 + 2 = 24
If the smallest digit to be placed in blank space = 0
Then the sum = 24 + 0 = 24 which is divisible by 3.
Next, if the greatest digit to be placed in blank space = 9
Then the sum = 24 + 9 = 33 which is divisible by 3.
Hence. the required smallest digit = 0 and greatest digit = 9
Question 6.
Write a digit in the blank space of each of the following numbers so that the number formed is divisible by 11 :
a) 92_389
Solution:
Sum of the digits at.odd places = 9 + 3 + 2 = 14
Sum of the digits at even places = 8 + ( ) + 9 = 17 + ( )
Difference = 17 + ( ) – 14 = 3 + ( )
If the given number divisible by 11, then 3 + ( ) = 11
∴ ( ) = 11 – 3 = 8
Hence, the required digit = 8
b) 8 _ 9484
Solution:
Sum of the digits at odd places = 4 + 4 + ( ) = 8 + ( )
Sum of the digits at even places = 8 + 9 + 8 = 25
Difference = 25 – [8 + ( )] = 17 – ( )
If 17 – ( ) = 11, then given number will be divisible by 11
∴ 17 – 11 = 6
So, the missing digit = 6