NCERT Solutions for Class 6 Maths Chapter 3 Playing With Numbers

NCERT
SOLUTIONS
CLASS - 6TH

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Class : 6th
Subject : Maths
Chapter : 3
Chapter Name : Playing With Numbers

Exercise 3.1

Q1 Write all the factors of the following numbers :
(a) 24
(b) 15
(c) 21
(d) 27
(e) 12
(f) 20
(g) 18
(h) 23
(i) 36

Answer. (a) 24
24 = 1 × 24, 24 = 2 × 12, 24 = 3 × 8

24 = 4 × 624 = 6 × 4

∴ Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24

(b) 15
15 = 1 × 15, 15 = 3 × 5, 15 = 5 × 3

∴ Factors of 15 are 1, 3, 5, and 15

(c) 21
21 = 1 × 21, 21 = 3 × 7, 21 = 7 × 3

∴ Factors of 21 are 1, 3, 7, and 21

(d) 27
27 = 1 × 27, 27 = 3 × 9, 27 = 9 × 3

∴ Factors of 27 are 1, 3, 9, and 27

(e) 12
12 = 1 × 12, 12 = 2 × 6, 12 = 3 × 4, 12 = 4 × 3

∴ Factors of 12 are 1, 2, 3, 4, 6, and 12

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(f) 20
20 = 1 × 20, 20 = 2 × 10, 20 = 4 × 5, 20 = 5 × 4

∴ Factors of 20 are 1, 2, 4, 5, 10, and 20

(g) 18
18 = 1 × 18, 18 = 2 × 9, 18 = 3 × 6, 18 = 6 × 3

∴ Factors of 18 are 1, 2, 3, 6, 9, and 18

(h) 23
23 = 1 × 23, 23 = 23 × 1

∴ Factors of 23 are 1 and 23

(i) 36
36 = 1 × 36, 36 = 2 × 18, 36 = 3 × 12, 36 = 4 × 9

36 = 6 × 6

Page : 50 , Block Name : Exercise 3.1

Q2 Write rst ve multiples of :
(a) 5
(b) 8
(c) 9

Answer. (a)
5 × 1 = 5, 5 × 2 = 10, 5 × 3 = 15, 5 × 4 = 20, 5 × 5 = 25

∴ The required multiples are 5, 10, 15, 20, and 25.

(b)
8 × 1 = 8, 8 × 2 = 16, 8 × 3 = 24, 8 × 4 = 32, 8 × 5 = 40

∴ The required multiples are 8, 16, 24, 32, and 40.

(c)
9 × 1 = 9, 9 × 2 = 18, 9 × 3 = 27, 9 × 4 = 36, 9 × 5 = 45

∴ The required multiples are 9, 18, 27, 36, and 45.

Page : 50 , Block Name : Exercise 3.1

Q3 Match the items in column 1 with the items in column 2

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Column 1 Column 2

(i) 35 (a) Multiple of 8

(ii) 15 (b) Multiple of 7

(iii) 16 (c) Multiple of 70

(iv) 20 (d) Factor of 30

(v) 35 (e) Factor of 50

(f) Multiple of 20

Answer.
Column 1 Column 2

(i) 35 (b) Multiple of 7

(ii) 15 (d) Factor of 30

(iii) 16 (a) Multiple of 8

(iv) 20 (d) Factor of 20

(v) 25 (e) Factor of 50

Page : 50 , Block Name : Exercise 3.1

Q4 Find all the multiples of 9 upto 100.

Answer.
9 × 1 = 9, 9 × 2 = 18, 9 × 3 = 27, 9 × 4 = 36, 9 × 5 = 45

9 × 6 = 54, 9 × 7 = 63, 9 × 8 = 72, 9 × 9 = 81, 9 × 10 = 90

9 × 11 = 99

Therefore, the multiples of 9 up to 100 are

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, and 99

Page : 51 , Block Name : Exercise 3.1

Exercise 3.2

Q1 What is the sum of any two (a) Odd numbers? (b) Even numbers?

Answer. (a)
The sum of two odd numbers is even.

e.g. 1 + 3 = 4

13 + 19 = 32

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(b)
The sum of two even numbers is even.

e.g. 2 + 4 = 6

10 + 18 = 28

Page : 53 , Block Name : Exercise 3.2

Q2 State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.

Answer. (a) False 3 + 5 + 7 = 15, i.e., odd
(b) True 3 + 5 + 6 = 14, i.e., even
(c) True 3 x 5 x 7 = 105, i.e., odd
(d) False 4 ÷ 2 = 2, i.e., even
(e) False 2 is a prime number and it is also even
(f) False 1 and the number itself are factors of the number
(g) False 2 + 3 - S, i.e., odd
(h) True
(i) False 2 is a prime number
(j) True 2 x 4 = 8, i.e., even

Page : 53 , Block Name : Exercise 3.2

Q3 The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find
such pairs of prime numbers upto 100.

Answer. 17,71
37,73
79,97

Page : 53 , Block Name : Exercise 3.2

Q4 Write down separately the prime and composite numbers less than 20.

Answer. Prime numbers less than 20 are

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2, 3, 5, 7, 11, 13, 17, 19
Composite numbers less than 20 are
4, 6, 8, 9, 10, 12, 14, 15, 16, 18

Page : 53 , Block Name : Exercise 3.2

Q5 What is the greatest prime number between 1 and 10?

Answer. prime numbers between 1 and 10 are 2, 3, 5, and 7. Among these numbers, 7 is the
greatest.

Page : 53 , Block Name : Exercise 3.2

Q6 Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18

Answer. (a) 44 = 37 +7
(b) 36 = 31 +5
(c) 24 = 19 +5
(d) 18 = 11 +7

Page : 53 , Block Name : Exercise 3.2

Q7 Give three pairs of prime numbers whose difference is 2. [Remark : Two prime numbers whose
difference is 2 are called twin primes].

Answer. 3,5
41,43
71,73

Page : 53 , Block Name : Exercise 3.2

Q8 Which of the following numbers are prime?
(a) 23
(b) 51
(c) 37
(d) 26

Answer. (a)
23, 23 = 1 × 23, 23 = 23 × 1

23 has only two factors, 1 and 23. Therefore, it is a prime number.

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(b)
51, 51 = 1 × 51, 51 = 3 × 17

51 has four factors, 1, 3, 17, 51. Therefore, it is not a prime number. It is a composite number.

(c) 37
It has only two factors, 1 and 37. Therefore, it is a prime number.

(d) 26
26 has four factors (1, 2, 13, 26). Therefore, it is not a prime number. It is a composite number.

Page : 53 , Block Name : Exercise 3.2

Q9 Write seven consecutive composite numbers less than 100 so that there is no prime number
between them.

Answer. Between 89 and 97, both of which are prime numbers, there are 7 composite numbers.
They are:
90, 91, 92, 93, 94, 95, 96

Numbers Factors

90, 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

91, 1, 7, 13, 91

92, 1, 2, 4, 23, 46, 92

93, 1, 3, 31, 93

94, 1, 2, 47, 95

95, 1, 5, 19, 95

96 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Page : 53 , Block Name : Exercise 3.2

Q10 Express each of the following numbers as the sum of three odd primes:
(a) 21
(b) 31
(c) 53
(d) 61

Answer. (a) 21 = 3 + 7 + 11
(b) 31 = 5 + 7 + 19
(c) 53 = 3 + 19 + 31
(d) 61 = 11 + 19 + 31

Page : 54 , Block Name : Exercise 3.2

Q11 Write ve pairs of prime numbers less than 20 whose sum is divisible by 5. (Hint : 3 + 7 = 10)

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Answer.
2 + 3 = 5

2 + 13 = 15

3 + 17 = 20

7 + 13 = 20

19 + 11 = 30

Page : 54 , Block Name : Exercise 3.2

Q12 Fill in the blanks :
(a) A number which has only two factors is called a ______.
(b) A number which has more than two factors is called a ______.
(c) 1 is neither ______ nor ______.
(d) The smallest prime number is ______.
(e) The smallest composite number is _____.
(f) The smallest even number is ______.

Answer. (a) prime number
(b) Composite number
(c) Prime number, Composite number
(d) 2
(e) 4
(f) 2

Page : 54 , Block Name : Exercise 3.2

Exercise 3.3

Q1 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):

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Answer.

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Page : 57 , Block Name : Exercise 3.3

Q2 Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:
(a) 572
(b) 726352
(c) 5500
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150

Answer. (a) 572
The last two digits are 72. Since 72 is divisible by 4, the given number is also divisible by 4. The last
three digits are 572. Since 572 is not divisible by 8, the given number is also not divisible by 8.

(b) 726352
The last two digits are 52. As 52 is divisible by 4, the given number is also divisible by 4. The last
three digits are 352. Since 352 is divisible by 8, the given number is also
divisible by 8

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(c) 5500
Since, last two digits are 00, It is divisible by 4. The last 3 digits are 500. Since divisible by 8. 5 is
not divisible by 8, the given number is not divisible by 8.

(d) 6000
Since the last 2 digits are 00, the given number is divisible by 4. Since the last 3 digits are 000, the
given number is divisible by 8.

(e) 12159
The last 2 digits are 59. Since 59 is not divisible by 4, the given number is also not divisible by 4.
The last 3 digits are 159. Since 159 is not divisible by 8, the given number is not divisible by 8.

(f) 14560
The last two digits are 84. Since 84 is divisible by 4, the given number is divisible by 4.
The last three digits are 084. Since 084 is not divisible by 8, the given number is not divisible by 8.

(h) 31795072
The last two digits are 72. Since 72 is divisible by 4, the given number is divisible by 4.
The last three digits are 072. Since 072 is divisible by 8, the given number is divisible by 8.

(i) 1700
The last two digits are 00. Since 00 is divisible by 4, the given number is divisible by 4. The last
three digits are 700. Since 700 is not divisible by 8, the given number is not
divisible by 8.

(j) 2150
The last two digits are 50. Since 50 is not divisible by 4, the given number is not
divisible by 4. The last three digits are 150. Since ISO is not divisible by 8, the given number is not
divisible by 8.

Page : 57 , Block Name : Exercise 3.3

Q3 Using divisibility tests, determine which of following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852

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Answer. (a) 297144
Since the last digit Of the number is 4, it is divisible by 2. On adding all the digits of the number,
the sum obtained is 27. Since 27 is divisible by 3, the given number is also divisible by 3.
As the number is divisible by both 2 and 3, it is divisible by 6.

(b) 1258
Since the last digit of the number is 8, it is divisible by 2. On adding all the digits of the number,
the sum obtained is 16. Since 16 is not divisible by 3, the given number is also not divisible by 3.
As the number is not divisible by both 2 and 3, it is not divisible by 6.

(c) The last digit of the number is 5, which is not divisible by 2. Therefore, the given number is also
not divisible by 2. On adding all the digits of the number, the sum obtained is 15. Since 15 is
divisible by 3, the given number is also divisible by 3.
As the number is not divisible by both 2 and 3, it is not divisible by 6.

(d) 61233
The last digit of the number is 3, which is not divisible by 2. Therefore, the given number is also
not divisible by 2. On adding all the digits Of the number, the sum obtained is 15. Since 15 is
divisible by 3, the given number is also divisible by 3.
As the number is not divisible by both 2 and 3, it is not divisible by 6.

(e) 901352
Since the last digit of the number is 2, it is divisible by 2. On adding all the digits of the number,
the sum obtained is 20. Since 20 is not divisible by 3, the given number is also not divisible by 3.
As the number is not divisible by both 2 and 3, it is not divisible by 6.

(f) 438750
Since the last digit of the number is O, it is divisible by 2. On adding all the digits of the number,
the sum obtained is 27. Since 27 is divisible by 3, the given number is also divisible by 3.
As the number is divisible by both 2 and 3, it is divisible by 6.

(g) 1790184
Since the last digit of the number is 4, it is divisible by 2. On adding all the digits of the number,
the sum obtained is 30. Since 30 is divisible by 3, the given number is also divisible by 3.
As the number is divisible by both 2 and 3, it is divisible by 6.

(h) 12583
Since the last digit of the number is 3, it is not divisible by 2. On adding all the digits of the
number, the sum obtained is 19. Since 19 is not divisible by 3, the given number is also not
divisible by 3.
As the number is not divisible by both 2 and 3, it is not divisible by 6.

(i) 639210
Since the last digit of the number is O, it is divisible by 2. On adding all the digits of the number,
the sum Obtained is 21. Since 21 is divisible by 3, the given number is also divisible by 3.

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As the number is divisible by both 2 and 3, it is divisible by 6.

(j) 17852
Since the last digit of the number is 2, it is divisible by 2. On adding all the digits of the number,
the sum obtained is 23. Since 23 is not divisible by 3, the given number is also not divisible by 3.
As the number is not divisible by both 2 and 3, it is not divisible by 6.

Page : 57 , Block Name : Exercise 3.3

Q4 Using divisibility tests, determine which of the following numbers are divisible by 11:
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153

Answer. (a) 5445
Sum of the digits at odd places = 5 + 4 = 9
Sum of the digits at even places = 4 + 5 = 9
Difference = 9 - 9 = 0
As the difference between the sum of the digits at odd places and the sum of the digits
at even places is O, therefore, 5445 is divisible by 11.

(b) 10824
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
The difference between the sum of the digits at odd places and the sum of the digits at
even places is 11, which is divisible by 11. Therefore, 10824 is divisible by 11.

(c) 7138965
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
The difference between the sum of the digits at odd places and the sum of digits at even places is 9,
which is not divisible by 11. Therefore, 7138965 is not divisible by 11.

(d) 70169308
Sum of the digits at odd places = 8 + 3 + 6 + 0
Sum of the digits at even places = 0 + 9 + 1 + 7 = 17
Difference = 17 - 17 = 0
As the difference between the sum of the digits at odd places and the sum of the digits
at even places is O, therefore, 70169308 is divisible by 11.

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(e) 10000001
Sum Of the digits at Odd places = 1
Sum of the digits at even places 1
Difference = 1 - 1 = 0
As the difference between the sum of the digits at odd places and the sum of the digits
at even places is O, therefore, 10000001 is divisible by 11.

(f) 901153
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
The difference between the sum of the digits at odd places and the sum of the digits at
even places is 11, which is divisible by 11. Therefore, 901153 is divisible by 11.

Page : 57 , Block Name : Exercise 3.3

Q5 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
(b) 4765_2

Answer. (a) _6724
Sum of the remaining digits = 19
To make the number divisible by 3, the sum of its digits should be divisible by 3.
The smallest multiple of 3 which comes after 19 is 21.
Therefore, smallest number = 21 - 19 - 2
Now , 2 + 3 + 3 = 8
If we put 8, then the sum of the digits will be 27 and as 27 is divisible by 3, the number
will also be divisible by 3.
Therefore, the largest number is 8.

(b) 4765_2
Sum of the remaining digits = 24
To make the number divisible by 3, the sum of its digits should be divisible by 3. As 24 is already
divisible by 3, the smallest number that can be placed here is 0.
Now, 0 + 3 = 3
3+3=6
3+3+3=9
How ever, 3 + 3 + 3 + 3 = 12
If we put 9, then the sum of the digits will be 33 and as 33 is divisible by 3, the number
will also be divisible by 3.
Therefore, the largest number is 9.

Page : 57 , Block Name : Exercise 3.3

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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
(b) 8 __ 9484

Answer. (a) 92_389
Let a be placed in the blank.
Sum of the digits at odd places = 9 + 3 + 2 = 14
Sum of the digits at even places = 8 + a + 9 = 17
Difference = 17 + a - 14 = 3 + a
For a number to be divisible by 11, this difference should be zero or a multiple of 11.
If 3 + a = O, then
a=-3
However, it cannot be negative.
A closest multiple of 11, which is near to 3, has to be taken. It is 11 itself.
3 + a = 11
a=8
Therefore, the required digit is 8.

(b) 8_9484
Let a be placed in the blank.
Sum Of the digits at odd places = 4 + 4 + a = 8 + a
Sum of the digits at even places = 8 + 9 + 8 = 25
Difference = 25 - (8 + a)
= 17 - a
For a number to be divisible by 11, this difference should be zero or a multiple of 11.
If 17 - a = 0, then
a = 17
This is not possible.
A multiple of 11 has to be taken. Taking 11, we obtain
17 - a = 11
a=6
Therefore, the required digit is 6.

Page : 58 , Block Name : Exercise 3.3

Exercise 3.4

Q1 Find the common factors of :
(a) 20 and 28
(b) 15 and 25
(c) 35 and 50

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(d) 56 and 120

Answer.
(a) Factors of 20 = 1, 2, 4, 5, 10, 20

Factors of 28 = 1, 2, 4, 7, 14, 28

Common factors = 1, 2, 4

(b) Factors of 15 = 1, 3, 5, 15

Factors of 25 = 1, 5, 25

Common factors = 1, 5

(c) Factors of 35 = 1, 5, 7, 35

Factors of 50 = 1, 2, 5, 10, 25, 50

Common factors = 1, 5

(d) Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56

Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Common factors = 1, 2, 4, 8

Page : 59 , Block Name : Exercise 3.4

Q2 Find the common factors of :
(a) 4, 8 and 12
(b) 5, 15 and 25

Answer.
(a) 4, 8, 12

Factors of 4 = 1, 2, 4

Factors of 8 = 1, 2, 4, 8

Factors of 12 = 1, 2, 3, 4, 6, 12

Common factors = 1, 2, 4

(b) 5, 15, and 25

Factors of 5 = 1, 5

Factors of 15 = 1, 3, 5, 15

Factors of 25 = 1, 5, 25

Common factors = 1, 5

Page : 59 , Block Name : Exercise 3.4

Q3 Find rst three common multiples of :
(a) 6 and 8

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(b) 12 and 18

Answer.
(a) 6 and 8

Multiple of 6 = 6, 12, 18, 24, 30 …

Multiple of 8 = 8, 16, 24, 32 … …

3 common multiples = 24, 48, 72

(b) 12 and 18

Multiples of 12 = 12, 24, 36, 78

Multiples of 18 = 18, 36, 54, 72

3 common multiples = 36, 72, 108

Page : 59 , Block Name : Exercise 3.4

Q4 Write all the numbers less than 100 which are common multiples of 3 and 4.

Answer.
Multiples of 3 = 3, 6, 9, 12, 15 …

Multiples of 4 = 4, 8, 12, 16, 20 …

Common multiples = 12, 24, 36, 48, 60, 72, 84, 96

Page : 59 , Block Name : Exercise 3.4

Q5 Which of the following numbers are co-prime?
(a) 18 and 35
(b) 15 and 37
(c) 30 and 415
(d) 17 and 68
(e) 216 and 215
(f) 81 and 16

Answer.
(a) Factors of 18 = 1, 2, 3, 6, 9, 18

Factors of 35 = 1, 5, 7, 35

Common factor = 1

Therefore, the given two numbers are co-prime.

(b) Factors of 15 = 1, 3, 5, 15

Factors of 37 = 1, 37

Common factors = 1

Therefore, the given two numbers are co-prime.

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(c) Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30

Factors of 415 = 1, 5, 83, 415

Common factors = 1, 5

As these numbers have a common factor other than 1, the given two numbers are nor c-prime.

(d) Factors of 17 = 1, 17

Factors of 68 = 1, 2, 4, 17, 34, 68

Common factors = 1, 17

As these numbers have a common factor other than 1, the given two numbers are nor c-prime.

(e) 216 and 215

Factors of 216 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216

Factors of 215 = 1, 5, 43, 215

Common factors = 1

Therefore, the given two numbers are co-prime.

(f) 81 and 16

Factors of 81 = 1, 3, 9, 27, 81

Factors of 16 = 1, 2, 4, 8, 16

Common factors = 1

Therefore, the given two numbers are co-prime

Page : 59 , Block Name : Exercise 3.4

Q6 A number is divisible by both 5 and 12. By which other number will that number be always
divisible?

Answer.
Factors of 5 = 1, 5

Factors of 12 = 1, 2, 3, 4, 6, 12

As the common factor of these numbers is 1, the given two numbers are co- prime and
the number will also be divisible by their product, i.e. 60, and the factors of 60, i.e., 1,
2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Page : 59 , Block Name : Exercise 3.4

Q7 A number is divisible by 12. By what other numbers will that number be divisible?

Answer. Since the number is divisible by 12, it will also be divisible by its factors i.e., 1, 2, 3, 4, 6,
12. Clearly, 1, 2, 3, 4, and 6 are numbers other than 12 by which this number is also divisible.

Page : 59 , Block Name : Exercise 3.4

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Exercise 3.5

Q1 Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers
separately.

Answer. (a) False
6 is divisible by 3, but not by 9.

(b) True, as 9 = 3 x 3
Therefore, if a number is divisible by 9, then it will also be divisible by 3.

(c) False
30 is divisible by 3 and 6 both, but it is not divisible by 18.

(d) True, as 9 x 10 = 90
Therefore, If a number is divisible by 9 and 10 both, then it will also be divisible by 90.

(e) False
15 and 32 are co-primes and also composite.

(f) False
12 is divisible by 4, but not by 8.

(g) True, as 8 = 2 x 4
Therefore if a number is divisible by 8, then it will also be divisible by 2 and 4.

(h) True
2 divides 4 and 8 as well as 12. (4 + 8 = 12)

(i) False
2 divides 12, but not divide 7 and 5.

Page : 61 , Block Name : Exercise 3.5

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Q2 Here are two different factor trees for 60. Write the missing numbers.

Answer. (a) As 6 = 2 x 3 and 10 = 5 x 2

(b) As 60 = 30 x 3, 30 = 10 x 3 and 10 = 5 x 2

Page : 62 , Block Name : Exercise 3.5

Q3 Which factors are not included in the prime factorisation of a composite number?

Answer. 1 and the number itself.

Page : 62 , Block Name : Exercise 3.5

Q4 Write the greatest 4-digit number and express it in terms of its prime factors.

Answer. Greatest four-digit number = 9999
9999 = 3 x 3 x 11 x 101

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Page : 62 , Block Name : Exercise 3.5

Q5 Write the smallest 5-digit number and express it in the form of its prime factors.

Answer. Smallest ve-digit number = 10,000
10000 = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5

Page : 62 , Block Name : Exercise 3.5

Q6 Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation,
if any; between two consecutive prime factors.

Answer.

1729 = 7 x 13 x 19
13 - 7 = 6, 19 - 13 = 6
The difference of two composite prime factors is 6.

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Page : 62 , Block Name : Exercise 3.5

Q7.The product of three consecutive numbers is always divisible by 6. Verify this statement with
the help of some examples.

Answer.
2 × 3 × 4 = 24, which is divisible by 6

9 × 10 × 11 = 990, which is divisible by 6

20 × 21 × 22 = 9240, which is divisible by 6

Page : 62 , Block Name : Exercise 3.5

Q8 The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help
of some examples.

Answer.
3 + 5 = 8, which is divisible by 4

15 + 17 = 32, which is divisible by 4

19 + 21 = 40, which is divisible by 4

Page : 62 , Block Name : Exercise 3.5

Q9 In which of the following expressions, prime factorisation has been done?
(a) 24 = 2 × 3 × 4
(b) 56 = 7 × 2 × 2 × 2
(c) 70 = 2 × 5 × 7
(d) 54 = 2 × 3 × 9

Answer.
(a) 24 = 2 × 3 × 4

since 4 is composite, prime factorisation has not been done.

(b) 56 = 7 × 2 × 2 × 2

since all the factors are prime, prime factorisation has been done.

(c) 70 = 2 × 5 × 7

since all the factors are prime, prime factorisation has been done.

(d) 54 = 2 × 3 × 9

since 9 is composite, prime factorisation has not been done.

Page : 62 , Block Name : Exercise 3.5

Q10 Determine if 25110 is divisible by 45.
[Hint : 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].

Page 21 of 31 Aglasem Schools

Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Answer. 45 = 5 x 9
Factors of 5 = 1, 5
Factors of 9 = 1. 3, 9
Therefore, 5 and 9 are co-prime numbers.
Since the last digit of 25110 is 0. it is divisible by 5.
Sum of the digits of 25110 = 2 + 5 + 1 +1 + 0 = 9
As the sum of the digits of 25110 is divisible by 9, therefore. 25110 is divisible by 9.
Since the number is divisible by 5 and 9 both, it is divisible by 45.

Page : 62 , Block Name : Exercise 3.5

Q11 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible
by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an
example to justify your answer.

Answer. No. It is not necessary because 12 and 36 are divisible by 4 and 6 both, but are not
divisible by 24.

Page : 62 , Block Name : Exercise 3.5

Q12 I am the smallest number, having four different prime factors. Can you nd me?

Answer. Since it is the smallest number of such type, it will be the product of 4 smallest prime
numbers.
2 x 3 x 5 x 7 = 210

Page : 62 , Block Name : Exercise 3.5

Exercise 3.6

Q1 Find the HCF of the following numbers :
(a) 18, 48
(b) 30, 42
(c) 18, 60
(d) 27, 63
(e) 36, 84
(f) 34, 102
(g) 70, 105, 175
(h) 91, 112, 49
(i) 18, 54, 81
(j) 12, 45, 75

Page 22 of 31 Aglasem Schools

Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Answer. (a) 18 = 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
HCF = 2 x 3 =6

(b) 30 = 2 x 3 x 5
42 = 2 x 3 x 7
HCF = 2 x 3 =6

(c) 18 = 2 x 3 x 3
60 = 2 x 2 x 3 x 5
HCF = 2 x 3 = 6

(d) 27 = 3 x 3 x 3
63 = 3 x 3 x 7
HCF = 3 x 3 = 9

(e) 36 = 2 x 3 x 3 x 3
84 = 2 x 2 x 3 x 7
HCF = 2 x 2 x 3 = 12

(f) 34 = 2 x 17
102 = 2 x 3 x 17
HCF = 2 x 17 = 34

(g) 70 = 2 x 5 x 7
105 = 3 x 5 x 7
HCF = 5 x 7 = 35

(h) 91 = 7 x 13
112 = 2 x 2 x 2 x 2 x 7
49 = 7 x 7
HCF = 7

(i) 18 = 2 x 3 x 3
112 = 2 x 3 x 3 x 3
81 = 3 x 3 x 3 x 3
HCF = 3 x 3 = 9

(j) 12 = 2 x 2 x 3
45 = 3 x 3 x 5
75 = 3 x 5 x 5
HCF = 2

Page : 63 , Block Name : Exercise 3.6

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Q2 What is the HCF of two consecutive
(a) numbers?
(b) even numbers?
(c) odd numbers?

Answer. (i) 1 e.g., HCF of 2 and 3 is 1.
(ii) 2 e.g., HCF of 2 and 4 is 2.
(iii) 1 e.g., HCF of 3 and 5 is 1.

Page : 63 , Block Name : Exercise 3.6

Q3 HCF of co-prime numbers 4 and 15 was found as follows by factorisation : 4 = 2 × 2 and 15 = 3 ×
5 since there is no common prime factor, so HCF of 4 and 15 is 0. Is the answer correct? If not,
what is the correct HCF?

Answer. No, the answer is not correct. 1 is the correct HCF.

Page : 64 , Block Name : Exercise 3.6

Exercise 3.7

Q1 Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of
weight which can measure the weight of the fertiliser exact number of times.

Answer. Weight of the two bags = 75 kg and 69 kg
Maximum weight - HCF (75, 69)
75 = 3 x 5 x 5
69 = 3 x 23
HCF = 3

Page : 67 , Block Name : Exercise 3.7

Q2 Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm
respectively. What is the minimum distance each should cover so that all can cover the distance in
complete steps?

Answer.
st
Step measure of 1 Boy = 63cm

nd
Step measure of 2 Boy = 70cm

rd
Step measure of 3 Boy = 77cm

LCM of 63, 70, 77

Page 24 of 31 Aglasem Schools

Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

LCM = 2 x 3 x 3 x 5 x 7 x 11=6930
Hence, the minimum distance each should cover so that all can cover the distance in
complete steps is 6930 cm.

Page : 67 , Block Name : Exercise 3.7

Q3 The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the
longest tape which can measure the three dimensions of the room exactly.

Answer.
Length = 825cm = 3 × 5 × 5 × 11

Breadth = 675cm = 3 × 3 × 3 × 5 × 5

Height = 450cm = 2 × 3 × 3 × 5 × 5

Longest tape = HCF of 825, 675, and 450 = 3 × 5 × 5 = 75cm

Therefore, the longest tape is 75cm.

Page : 67 , Block Name : Exercise 3.7

Q4 Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.

Answer. Smallest number = LCM of 6, 8, 12

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

We have to nd the smallest 3-digit multiple of 24.
It can be seen that 24 x 4 = 96 and 24 x 5 = 120.
Hence, the smallest 3-digit number which is exactly divisible by 6, 8, and 12 is 120.

Page : 67 , Block Name : Exercise 3.7

Q5 Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.

Answer. LCM of 8, 10 and 12

LCM = 2 x 2 x 2 x 3 x 5 = 120
We have to nd the greatest 3-digit multiple of 120.
It can be seen that 120 x 8 = 960 and 120 x 9 = 1080.
Hence, the greatest 3-digit number exactly divisible by 8, 10, and 12 is 960.

Page : 67 , Block Name : Exercise 3.7

Q6 The traf c lights at three different road crossings change after every 48 seconds, 72 seconds
and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they
change simultaneously again?

Answer. Time period after which these lights will change = LCM of 48, 72 and 108

Page 26 of 31 Aglasem Schools

Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

LCM=2 x 2 x 2 x 2 x 3 x 3 x 3 = 432
They will change together after every 432 seconds i.e., 7 min 12 seconds.
Hence, they will change simultaneously at 7:07:12 am.

Page : 67 , Block Name : Exercise 3.7

Q7 Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the
maximum capacity of a container that can measure the diesel of the three containers exact
number of times.

Answer.
Maximum capacity of the required tanker = HCF of 403, 434, 465

403 = 13 × 31

434 = 2 × 7 × 31

465 = 3 × 5 × 31

HCF = 31

∴ A container of capacity 31 can measure the diesel of 3 containers exact number of times.

Page : 67 , Block Name : Exercise 3.7

Q8 Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.

Answer. LCM of 6, 15, 18

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

LCM = 2 x 2 x 3 x 5 = 90
Required number = 90 + 5 = 95

Page : 67 , Block Name : Exercise 3.7

Q9 Find the smallest 4-digit number which is divisible by 18, 24 and 32.

Answer. LCM of 18, 24 and 32

LCM = 2 x 2 x 2 x 2 x 2 x 3 x 3
We have to nd the smallest 4-digit multiple of 288.
It can be observed that 288 x 3 = 864 and 288 x 4 = 1152.
Therefore, the smallest 4-digit number which is divisible by 18, 24, and 32 is 1152.

Page : 67 , Block Name : Exercise 3.7

Q10 Find the LCM of the following numbers :
(a) 9 and 4
(b) 12 and 5
(c) 6 and 5

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

(d) 15 and 4
Observe a common property in the obtained LCMs. Is LCM the product of two numbers in each
case?

Answer. (a) LCM = 2 x 2 x 3 x 3 = 36

(b) LCM = 2 x 2 x 3 x 5 = 60

(c) LCM = 2 x 3 x 4 = 30

(d) LCM = 2 x 3 x 5 = 30

Page 29 of 31 Aglasem Schools

Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

Yes, it can be observed that in each case, the LCM of the given numbers is the product
of these numbers. When two numbers are co-prime, their LCM is the product of those
numbers. Also, in each case, LCM is a multiple of 3.

Page : 67 , Block Name : Exercise 3.7

Q11 Find the LCM of the following numbers in which one number is the factor of the other.
(a) 5, 20
(b) 6, 18
(c) 12, 48
(d) 9, 45
What do you observe in the results obtained?

Answer. (a) LCM = 2 x 2 x 5 = 20

(b) LCM = 2 x 3 x 3 = 18

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Book : Mathematics Ncert Solutions | Chapter- 3 Mathematics

(c) LCM = 2 x 2 x 2 x 2 x 3 = 48

(d) LCM = 3 x 3 x 5 = 45

Yes, it can be observed that in each case, the LCM of the given numbers is the larger
number. When one number is a factor of the other number, their LCM will be the larger
number.

Page : 67 , Block Name : Exercise 3.7

Page 31 of 31 Aglasem Schools