**PSEB Class 6 Maths Chapter 3 Playing With Numbers Questions and Answers Solution**

PSEB Punjab Board Class 6 Maths Textbook Solution Chapter 3 Playing With Numbers Exercise Questions and Answers. We have also included some Additional questions, so that students can prepare more strongly.

**Exercise 3.1**

**1.) Write down all the factors of each of the following:-**

(i) 18

ANSWER:

We have to write all the factors of 18.

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

(ii) 24

ANSWER:

We have to write all the factors of 24

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

(iii) 45

ANSWER:

We have to write all the factors of 45

Factors of 45 are 1, 5, 9 and 45

(iv) 60

ANSWER:

We have to write all the factors of 60

Factors of 60 are 1,2,3,4,5,6,10,12,15,20,30 and 60

(v) 65

ANSWER:

We have to write all the factors of 65

Factors of 65 are 1,5,13 and 65

**2.) Write down the first six multiples of each of the following:-**

(i) 6

ANSWER:

We have to write first six multiples of 6

First six multiples of 6 are 6, 12,18,24,30 and 36

(ii) 9

ANSWER:

We have to write first six multiples of 9

First six multiples of 9 are 9, 18,27,36,45 and 54

(iii) 11

ANSWER:

We have to write first six multiples of 11.

First six multiples of 11 are 11, 22,33,44,55 and 66

(iv) 15

ANSWER:

We have to write first six multiples of 15.

First six multiples of 15 are 15, 30,45,60,75 and 90

(v) 24

ANSWER:

We have to write first six multiples of 24

First six multiples of 24 are 24,48,72,96,120 and 144

**3.) List all the numbers less than 100 that are multiples of**

(i) 17

ANSWER:

We have to write all the numbers less than 100 that are multiples of 17

The numbers less than 100 that are multiples of 17 are 17,34,51,68 and 85

(ii) 12

ANSWER:

We have to write all the numbers less than 100 that are multiples of 12

The numbers less than 100 that are multiples of 12 are 12,24,36,48,60,72,84 and 96

(ii) 21

ANSWER:

We have to write all the numbers less than 100 that are multiples of 21

The numbers less than 100 that are multiples of 21 are 21, 42, 63 and 84

4.) Which of the following are prime numbers?

(i) 39

ANSWER:

We know,

Numbers whose only factors are 1 and the number itself are called prime numbers.

39 has more than 2 factors.

**39 is not prime number.**

(ii) 127

ANSWER:

We know,

Numbers whose only factors are 1 and the number itself are called prime numbers.

127 has only 2 factor i.e. 1 and number itself

**127 is prime number.**

** **

(iii) 177

ANSWER:

We know,

Numbers whose only factors are 1 and the number itself are called prime numbers.

177 has more than 2 factors.

**177 is not prime number.**

(iv) 201

ANSWER:

We know,

Numbers whose only factors are 1 and the number itself are called prime numbers.

201 has more than 2 factors.

**201 is not prime number.**

(v) 237

ANSWER:

We know,

Numbers whose only factors are 1 and the number itself are called prime numbers.

237 has more than 2 factors.

**237 is not prime number.**

** **

(vi) 361

ANSWER:

We know,

Numbers whose only factors are 1 and the number itself are called prime numbers.

361 has more than 2 factors.

**361 is not prime number.**

**5.) Express each of the following as sum of two odd prime numbers:-**

(i) 16

ANSWER:

We have to express 16 as sum of two odd prime numbers.

**16 = 3 + 13**

**16 = 5 + 11**

(ii) 28

ANSWER:

We have to express 28 as sum of two odd prime numbers.

**28 = 5 + 23**

**28 = 11 + 17**

(iii) 40

ANSWER:

We have to express 40 as sum of two odd prime numbers.

**40 = 3 + 37**

**40 = 11 + 29**

**40 = 17 + 23**

**6.) Write all the prime numbers between the given numbers:-**

(i) 1 to 25

ANSWER:

We have to write prime numbers between 1 to 25

Prime numbers between 1 to 25 = 2, 3, 5,7,11,13,17,19 and 23

(ii) 85 to 105

ANSWER:

We have to write prime numbers between 85 to 105

Prime numbers between 85 to 105 = 89, 97,101 and 103

(iii) 120 to 140

ANSWER:

We have to write prime numbers between 120 to 140

Prime numbers between120 to 140 = 127,131,137 and 139

**7.) Is 36 a perfect number?**

ANSWER:

We know,

Perfect number is the sum of all the factors of a number is 2 times of the given number.

Factors of 36 are 1,2,3,4,6,9,12,18 and 36

Sum of Factors of 36 = 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36

Sum of Factors of 36 = 91

Sum of Factors of 36 is not 2 times of 36

**36 is not a perfect number.**

**8.) Find the missing factors:-**

(i) 5x….. = 30

ANSWER:

We have to find missing factors.

Missing factor = 30 / 5

**Missing factor = 6**

(ii)….. x6 = 48

ANSWER:

We have to find missing factors.

Missing factor = 48 / 6

**Missing factor = 8**

** **

(iii) 7 x…..= 63

ANSWER:

We have to find missing factors.

Missing factor = 63 / 7

**Missing factor = 9**

** **

(iv)—x8 = 104

ANSWER:

We have to find missing factors.

Missing factor = 104 / 8

**Missing factor = 13**

(v)—– x 7 = 105

ANSWER:

We have to find missing factors.

Missing factor = 105 / 7

**Missing factor = 15**

**9.) List all 2-digit prime numbers, in which both the digits are prime numbers.**

ANSWER:

We have to list all 2-digit prime numbers, in which both the digits are prime numbers.

**The 2-digit prime numbers are 23, 37, 53 and 73**

**Exercise 3.2**

**1.) Find the common factors of the followings:-**

** (i) 16 and 24**

ANSWER:

We have to find common factors of 16 and 24.

We 1^{st} find factors of given numbers.

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

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

**Common factors of 16 and 24 are 1,2,4,8**

(ii) 25 and 40

ANSWER:

We have to find common factors of 25 and 40.

We 1st find factors of given numbers.

Factors of 25 = 1, 5 and 25

Factors of 40 = 1, 2, 4,5,8,10,20 and 40

**Common factors of 25 and 40 are 1 and 5**

(iii) 24 and 36

ANSWER:

We have to find common factors of 24 and 36.

We 1st find factors of given numbers.

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

Factors of 36 = 1, 2, 3, 4,6,9,12,18 and 36

**Common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12**

(iv) 14, 35 and 42

ANSWER:

We have to find common factors of 14, 35 and 42.

We 1st find factors of given numbers.

Factors of 14 = 1, 2, 7 and 14

Factors of 35 = 1, 5, 7 and 35

Factors of 42 = 1, 2, 3,6,7,14,21 and 42

**Common factors of 14, 35 and 42 are 1 and 7.**

(v) 15, 24 and 35

ANSWER:

We have to find common factors of 15, 24 and 35

We 1st find factors of given numbers.

Factors of 15 = 1, 3, 5 and 15

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

Factors of 35 = 1, 5, 7 and 35

**Common factors of 15, 24 and 35 are 1.**

**2.) Find first three common multiples of the followings:-**

(i) 3 and 5

ANSWER:

We have to find first three common multiples of 3 and 5

We 1st find multiples of 3 and 5

Multiples of 3 = 3, 6, 9,12,15,18,21,24,27,30,33,36,39,42,45

Multiples of 5 = 5, 10, 15,20,25,30,35,40,45

**First three common multiples of 3 and 5 = 15, 30 and 45**

(ii) 6 and 8

ANSWER:

We have to find first three common multiples of 6 and 8

We 1st find multiples of 6 and 8

Multiples of 6 = 6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96

Multiples of 8 = 8,16,24,32,40,48,56,64,72,80,88,96

**First three common multiples of 6 and 8 = 24, 48 and 72**

** **

(iii) 2, 3 and 4

ANSWER:

We have to find first three common multiples of 2, 3 and 4

We 1st find multiples of 2, 3 and 4

Multiples of 2 = 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40

Multiples of 3 = 3, 6,9,12,15,18,21,24,27,30

Multiples of 4 = 4, 8, 12, 16,20,24,28,32,36,40

**First three common multiples of 2, 3 and 4 are 12, 24 and 36**

**3.) Which of the following numbers are divisible by 2 or 4?**

(i) 52314

ANSWER:

We know,

If the One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

A number is divisible by 4, if the number formed by its last two digits (i.e. tens and ones) is divisible by 4.

**52314 = One’s digit of a number is 4, the number is divisible by 2.**

**52314 = last two digits 14 not divisible by 4, number is not divisible by 4**

(ii) 678913

ANSWER:

We know,

If the One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

A number is divisible by 4, if the number formed by its last two digits (i.e. tens and ones) is divisible by 4.

**678913 = One’s digit of a number is 3, the number is not divisible by 2.**

**678913 = last two digits 13 not divisible by 4, number is not divisible by 4**

** **

(ⅲ) 4056784

ANSWER:

We know,

If the One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

A number is divisible by 4, if the number formed by its last two digits (i.e. tens and ones) is divisible by 4.

**4056784 = One’s digit of a number is 4, the number is divisible by 2.**

**4056784 = last two digits 84 divisible by 4, number is divisible by 4.**

** **

(iv) 21536

ANSWER:

We know,

If the One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

**21536 = One’s digit of a number is 6, the number is divisible by 2.**

**21536 = last two digits 36 divisible by 4, number is divisible by 4.**

(v) 412318

ANSWER:

We know,

If the One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

**412318 = One’s digit of a number is 8, the number is divisible by 2.**

**412318 = last two digits 18 not divisible by 4, number is not divisible by 4**

**4.) Which of the following numbers are divisible by 3 or 9?**

(i) 654312

ANSWER:

We know,

If sum of digits of a number is divisible by 3 then the number is divisible by 3.

If sum of digits of a number is divisible by 9 then the number is divisible by 9.

654312 = 6 + 5 + 4 + 3 + 1 + 2

**6 + 5 + 4 + 3 + 1 + 2 = 21**

21 is divisible by 3 then the number is divisible by 3.

21 is not divisible by 9 then the number is not divisible by 9.

**654312 is divisible by 3 only.**

(ii) 516735

ANSWER:

We know,

If sum of digits of a number is divisible by 3 then the number is divisible by 3.

If sum of digits of a number is divisible by 9 then the number is divisible by 9.

516735 = 5 + 1 + 6 + 7 + 3 + 5

5 + 1 + 6 + 7 + 3 + 5 = 27

27 is divisible by 3 then the number is divisible by 3.

27 is divisible by 9 then the number is divisible by 9.

**516735 is divisible by 3 and 9 both.**

(iii) 423152

ANSWER:

We know,

If sum of digits of a number is divisible by 3 then the number is divisible by 3.

If sum of digits of a number is divisible by 9 then the number is divisible by 9.

423152 = 4 + 2 + 3 + 1 + 5 + 2

4 + 2 + 3 + 1 + 5 + 2 = 17

17 is not divisible by 3 then the number is not divisible by 3.

17 is not divisible by 9 then the number is not divisible by 9.

**423152 is not divisible by 3 and 9. **

(iv) 704355

ANSWER:

We know,

If sum of digits of a number is divisible by 3 then the number is divisible by 3.

If sum of digits of a number is divisible by 9 then the number is divisible by 9.

704355 = 7 + 0 + 4 + 3 + 5 + 5

7 + 0 + 4 + 3 + 5 + 5 = 24

24 is divisible by 3 then the number is divisible by 3.

24 is not divisible by 9 then the number is not divisible by 9.

**704355 is divisible by 3 only.**

** **

(v) 215478

ANSWER:

We know,

If sum of digits of a number is divisible by 3 then the number is divisible by 3.

If sum of digits of a number is divisible by 9 then the number is divisible by 9.

215478 = 2 + 1 + 5 + 4 + 7 + 8

2 + 1 + 5 + 4 + 7 + 8 = 27

27 is divisible by 3 then the number is divisible by 3.

27 is divisible by 9 then the number is divisible by 9.

**215478 is divisible by 3 and 9 both.**

** **

**5.) Which of the following numbers are divisible by 5 or 10?**

(i) 456803

ANSWER:

We know,

If the One’s digit of a number is 0 or 5, the number is divisible by 5.

If the One’s digit of a number is 0, the number is divisible by 10.

**456803 = One’s digit of a number is not 0 or 5, the number is not divisible by 5 and 10.**

** **

(ii) 654130

ANSWER:

We know,

If the One’s digit of a number is 0 or 5, the number is divisible by 5.

If the One’s digit of a number is 0, the number is divisible by 10.

**654130 = One’s digit of a number is 0, the number is divisible by 5 and 10**

** **

(iii) 256785

ANSWER:

We know,

If the One’s digit of a number is 0 or 5, the number is divisible by 5.

If the One’s digit of a number is 0, the number is divisible by 10.

256785 = One’s digit of a number is 5, the number is divisible by 5.

256785 = One’s digit of a number is not 0, the number is not divisible by 10.

**256785 is divisible by 5 only.**

(iv) 412508

ANSWER:

We know,

If the One’s digit of a number is 0 or 5, the number is divisible by 5.

If the One’s digit of a number is 0, the number is divisible by 10.

**412508 = One’s digit of a number is not 0 or 5, the number is not divisible by 5 and 10.**

** **

(v) 872565

ANSWER:

We know,

If the One’s digit of a number is 0 or 5, the number is divisible by 5.

If the One’s digit of a number is 0, the number is divisible by 10.

872565 = One’s digit of a number is 5, the number is divisible by 5.

872565 = One’s digit of a number is not 0, the number is not divisible by 10.

**872565 is divisible by 5 only.**

**6.) Which of the following numbers are divisible by 8?**

(i) 457432

ANSWER:

We know,

A number with 4 or more digits is divisible by 8, if the number formed by its last three digits is divisible by 8.

457432 = last three digits 432 is divisible by 8.

**457432 is divisible by 8.**

(ii) 5134214

ANSWER:

We know,

A number with 4 or more digits is divisible by 8, if the number formed by its last three digits is divisible by 8.

5134214 = last three digits 214 is not divisible by 8.

**5134214 is not divisible by 8.**

(iii) 7232000

ANSWER:

We know,

A number with 4 or more digits is divisible by 8, if the number formed by its last three digits is 000 then the number is divisible by 8.

7232000 = last three digits is 000

**7232000 is divisible by 8.**

(iv) 5124328

ANSWER:

We know,

A number with 4 or more digits is divisible by 8, if the number formed by its last three digits is divisible by 8.

5124328 = last three digits 328 is divisible by 8.

**5124328 is divisible by 8.**

** **

(v) 642516

ANSWER:

We know,

642516 = last three digits 516 is not divisible by 8.

**642516 is not divisible by 8.**

** **

**7.) Which of the following numbers are divisible by 6?**

(i) 425424

ANSWER:

If a number is divisible by both 2 and 3 then it is also divisible by 6.

The One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

Sum of digits of a number is divisible by 3 then the number is divisible by 3.

425424 = One’s digit of a number is even, number is divisible by 2.

425424 = 4 + 2 + 5 + 4 + 2 + 4 = 21 divisible by 3 then the number is divisible by 3.

**425424 is divisible by both 2 and 3 then it is also divisible by 6.**

** **

(ii) 617415

ANSWER:

If a number is divisible by both 2 and 3 then it is also divisible by 6.

The One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

Sum of digits of a number is divisible by 3 then the number is divisible by 3.

617415 = One’s digit of a number is not even, number is not divisible by 2.

617415 = 6 + 1 + 7 + 4 + 1 + 5 = 24 divisible by 3 then the number is divisible by 3.

**617415 is divisible by 3 then it is not divisible by 6.**

** **

(ⅲ) 3415026

ANSWER:

If a number is divisible by both 2 and 3 then it is also divisible by 6.

The One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

Sum of digits of a number is divisible by 3 then the number is divisible by 3.

3415026 = One’s digit of a number is even, number is divisible by 2.

3415026 = 3 + 4 + 1 + 5 + 0 + 2 + 6 = 21 divisible by 3 then the number is divisible by 3.

**3415026 is divisible by both 2 and 3 then it is also divisible by 6.**

** **

(iv) 4065842

ANSWER:

If a number is divisible by both 2 and 3 then it is also divisible by 6.

The One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

Sum of digits of a number is divisible by 3 then the number is divisible by 3.

4065842 = One’s digit of a number is even, number is divisible by 2.

4065842 =4 + 0 + 6 + 5 + 8 + 4 + 2 = 29 not divisible by 3 then the number is not divisible by 3.

**4065842 is divisible by 2 then it is not divisible by 6.**

** **

(v) 725436

ANSWER:

If a number is divisible by both 2 and 3 then it is also divisible by 6.

The One’s digit of a number is 0,2,4,6, or 8, the number is divisible by 2.

Sum of digits of a number is divisible by 3 then the number is divisible by 3.

725436 = One’s digit of a number is even, number is divisible by 2.

725436 = 7+ 2 + 5 + 4 + 3 + 6 = 27 divisible by 3 then the number is divisible by 3.

**725436 is divisible by both 2 and 3 then it is also divisible by 6.**

** **

**8.) Which of the following numbers are divisible by 11?**

(i) 4281970

ANSWER:

We know,

A given number is divisible by 11, if the difference between the sum of the digits at odd places and the sum of the digits at even places (from the right) is either 0 or divisible by 11.

4281970

Sum of the digits at odd places = 0 + 9 + 8 + 4 = 21

Sum of the digits at even places = 7 + 1 + 2 = 10

Difference = 21 – 10 = 11

**4281970 is divisible by 11.**

** **

(ii) 8049536

ANSWER:

We know,

A given number is divisible by 11, if the difference between the sum of the digits at odd places and the sum of the digits at even places (from the right) is either 0 or divisible by 11.

8049536

Sum of the digits at odd places = 6 + 5 + 4 + 8 = 23

Sum of the digits at even places = 3 + 9 + 0 = 12

Difference = 23 – 12 = 11

**8049536 is divisible by 11.**

** **

(iii) 1234321

ANSWER:

We know,

A given number is divisible by 11, if the difference between the sum of the digits at odd places and the sum of the digits at even places (from the right) is either 0 or divisible by 11.

1234321

Sum of the digits at odd places = 1 + 3 + 3 + 1 = 8

Sum of the digits at even places = 2 + 4 + 2 = 8

Difference = 8 – 8 = 0

**1234321 is divisible by 11.**

** **

(iv) 6450828

ANSWER:

We know,

6450828

Sum of the digits at odd places = 8 + 8 + 5 + 6 = 27

Sum of the digits at even places = 2 + 0 + 4 = 6

Difference = 27 – 6 = 21

**6450828 is not divisible by 11.**

** **

(v) 5648346

ANSWER:

We know,

5648346

Sum of the digits at odd places = 6 + 3 + 4 + 5 = 18

Sum of the digits at even places = 4 + 8 + 6 = 18

Difference = 18 – 18 = 0

**5648346 is divisible by 11.**

** **

**9.) State True or False:-**

(i) If a number is divisible by 24. Then it is also divisible by 3 and 8.

ANSWER:

True.

A number is divisible by 24. Then it is also divisible by 3 and 8.

**24 = 8 x 3**

** **

(ii) 60 and 90 both are divisible by 10 then their sum is not divisible by 10.

ANSWER:

False.

**60 and 90 both are divisible by 10 then their sum is also divisible by 10.**

(iii) If a number is divisible by 8 then it is also divisible by 16.

ANSWER:

False.

A number is divisible by 8 then it is will be or not will be divisible by 16.

(iv) If a number is divisible by 15 then it is also divisible by 3.

ANSWER:

True.

A number is divisible by 15 then it is also divisible by 3.

**15 = 3 x 5**

(v) 144 and 72 are divisible by 12 then their difference is also divisible by 12.

ANSWER:

True.

144 and 72 are divisible by 12 then their difference is also divisible by 12.

**10.) If a number is divisible by 5 and 9 then by which other number will that number be always divisible?**

ANSWER:

Given that, a number is divisible by 5 and 9

We have to find by which other number that number will be always divisible

Number will be always divisible by other number = Multiplication of given numbers

Number will be always divisible by other number = 5 x 9

**Number will be always divisible by other number = 45**

**11.) Which of the following pairs are co-prime?**

(i) 25, 35

ANSWER:

We know,

The numbers which have only 1 as the common factor are called co-primes.

In 25, 35 more than 1 common factors.

**25, 35 pair is not co-prime.**

(ii) 16, 21

ANSWER:

We know,

The numbers which have only 1 as the common factor are called co-primes.

In 16, 21 only 1 common factors.

**16, 21 pair is co-prime.**

(iii) 24, 41

ANSWER:

We know,

The numbers which have only 1 as the common factor are called co-primes.

In 24, 41 only 1 common factors.

**24, 41 pair is co-prime.**

** **

(iv) 48, 33

ANSWER:

We know,

The numbers which have only 1 as the common factor are called co-primes.

In 48, 33 more than 1 common factors.

**48, 33 pair is not co-prime.**

** **

(v) 20,57

ANSWER:

We know,

The numbers which have only 1 as the common factor are called co-primes.

In 20,57 only 1 common factors.

**20,57 pair is co-prime.**

** **

**Exercise 3.3**

**1.) Find prime factors of the following numbers by factor tree method:-**

(i) 96

ANSWER:

We have to do prime factorisation of 96 by factor tree method.

(ii) 120

ANSWER:

We have to do prime factorisation of 120 by factor tree method.

(iii) 180

ANSWER:

We have to do prime factorisation of 180 by factor tree method.

**2.) Complete each factor tree:-**

ANSWER:

We have to complete given factor tree.

ANSWER:

We have to complete given factor tree.

ANSWER:

We have to complete given factor tree.

**3.) Find the prime factors of the following numbers by division method:-**

(i) 420

ANSWER:

We have to do prime factors of 420 by division method

**Prime factors of 420 = 2 x 2 x 3 x 5 x 7**

(ii) 980

ANSWER:

We have to do prime factors of 980 by division method

**Prime factors of 980 = 2 x 2 x 5 x 7 x 7**

** **

(iii) 225

ANSWER:

We have to do prime factors of 225 by division method.

**Prime factors of 225 = 3 x 3 x 5 x 5**

** **

(iv) 150

ANSWER:

We have to do prime factors of 150 by division method.

**Prime factors of 150 = 2 x 3 x 5 x 5**

** **

(v) 324

ANSWER:

We have to do prime factors of 324 by division method.

**Prime factors of 324 = 2 x 2 x 3 x 3 x 3 x 3**

** **

**Exercise 3.4**

**1.) Find H.C.F. of the following numbers by prime factorisation:-**

(1) 30, 42

ANSWER:

We know,

The Highest Common Factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors.

We have to find the HCF of the 30, 42

**HCF of 30, 42 = 2 x 3 = 6**

** **

(ii) 135, 225

ANSWER:

We know,

The Highest Common Factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors.

We have to find the HCF of the 135, 225

**HCF of 135, 225 = 3 x 3 x 5 = 45**

** **

(ii) 180, 192

ANSWER:

We know,

The Highest Common Factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors.

We have to find the HCF of the 180, 192

**HCF of 180, 192 = 2 x 2 x 3 = 12**

(iv) 49, 91, 175

ANSWER:

We know,

We have to find the HCF of the 49, 91, and 175

**HCF of 49, 91, and 175 = 7**

** **

(v) 144, 252, 630

ANSWER:

We know,

We have to find the HCF of the 144, 252, and 630

**HCF of the 144, 252, and 630 = 2 x 3 x 3 = 18**

** **

**2.) Find H.C.F. of the following numbers using division method:-**

(i) 170, 238

ANSWER:

We have to find H.C.F. of 170, 238 using division method.

H.C.F. of 170, 238 using division method:

170) 238 (1

– 170

Remainder 68) 170 (2

– 136

Remainder 34) 136 (4

– 136

——————————– 00

**HCF of 170, 238 is 34.**

** **

(ii) 54, 144

ANSWER:

We have to find H.C.F. of 54, 144 using division method.

H.C.F. of 54, 144 using division method:

54) 144 (2

– 108

Remainder 18) 108 (6

– 108

——————————– 00

**H.C.F. of 54, 144 is 18.**

** **

(ⅲ) 72, 88

ANSWER:

We have to find H.C.F. of 72, 88 using division method.

H.C.F. of 72, 88 using division method:

72) 88 (1

– 72

Remainder 16) 72 (4

– 64

Remainder 8) 64 (8

– 64

——————————– 00

**H.C.F. of 72, 88 is 8.**

** **

(iv) 96, 240, 336

ANSWER:

We have to find H.C.F. of 96, 240, 336 using division method.

H.C.F. of 96, 240, 336 using division method:

We find 1^{st} H.C.F. of 96, 240.

96) 240 (2

– 192

Remainder 48) 192 (4

– 192

——————————– 00

H.C.F. of 96, 240 is 48.

Now,

We find HCF of the third number and the HCF of first two numbers.

48 and 336

48) 336 (7

– 336

——————————– 00

**H.C.F. of 96, 240, 336 is 48.**

** **

(v) 120, 156, 192

ANSWER:

We have to find H.C.F. of 120, 156, 192 using division method.

H.C.F. of 120, 156, 192 using division method:

We find 1st H.C.F. of 120, 156

120) 156 (1

– 120

Remainder 36) 120 (3

– 108

Remainder 12) 108 (9

– 108

——————————– 00

H.C.F. of 120, 156 is 12.

Now,

We find HCF of the third number and the HCF of first two numbers.

12 and 192

12) 192 (16

– 192

——————————– 00

**H.C.F. of 120, 156, and 192 is 12.**

3.) What is the HC.F.of two prime numbers?

ANSWER:

**HC.F. of two prime numbers is always 1.**

** **

4.) What is the H.C.F. of two consecutive even numbers?

ANSWER:

**H.C.F. of two consecutive even numbers is always 2.**

5.) What is the H.C.F of two consecutive natural numbers?

ANSWER:

**H.C.F of two consecutive natural numbers is always 1.**

** **

6.) What is the H.C.F. of two consecutive odd numbers?

ANSWER:

**The H.C.F. of two consecutive odd numbers is always 1.**

** **

7.) Find the greatest number which divides 245 and 1029, leaving a remainder 5 in each case.

ANSWER:

Here, we have to find HCF which divides 245 and 1029, leaving a remainder 5 in each case.

245 – 5 = 240

1029– 5 = 1024

We have to find HCF of 240, 1024

**HCF of 240, 1024 = 2 x 2 x 2 x 2 = 16**

**8.) Find the greatest number that can divide 782 and 460 leaving remainder 2 and 5 respectively**.

ANSWER:

Here, we have to find HCF that can divide 782 and 460 leaving remainder 2 and 5 respectively

782 – 2 = 780

460 – 5 = 455

We have to find HCF of 780 and 455

**HCF of 780 and 455 = 13 x 5 = 65**

**9.) Find the greatest number that will divide 398, 437 and 540 leaving remainders 7, 12 and 13 respectively.**

ANSWER:

Here, we have to find HCF that will divide 398, 437 and 540 leaving remainders 7, 12 and 13 respectively.

398 – 7 = 391

437 – 12 = 425

540 – 13 = 527

We have to find HCF of 391, 425 and 527

**HCF of 391, 425 and 527 is 17.**

**10.) Two different containers contain 529 litres and 667 litres of milk respectively. Find the maximum capacity of container which can measure the milk of both containers in exact number of times.**

ANSWER:

Given that,

Two different containers contain 529 litres and 667 litres of milk respectively

We have to find the maximum capacity of container which can measure the milk of both containers in exact number of times.

We have to find HCF of 529 litres and 667 litres.

**HCF of 529 litres and 667 litres is 23 litres.**

**The maximum capacity of container which can measure the milk of both containers in exact number of times is 23 litres.**

** **

**11.) There are 136 apples, 170 mangoes and 255 oranges. These are to be packed in boxes containing the same number of fruits. Find the greatest number of fruits possible in each box.**

ANSWER:

Given that,

There are 136 apples, 170 mangoes and 255 oranges.

We have to find the greatest number of fruits possible in each box.

We have to find HCF of 136 apples, 170 mangoes and 255 oranges.

HCF of 136 apples, 170 mangoes and 255 oranges is 17

**Greatest number of fruits possible in each box is 17.**

**12.) Three pieces of timber 54m, 36m and 24m long, have to be divided into planks of the same length. What is the greatest possible length of each plank?**

ANSWER:

Given that,

Three pieces of timber 54m, 36m and 24m long.

We have to find the greatest possible length of each plank.

We have to find HCF of 54m, 36m and 24m long.

HCF of 54m, 36m and 24m long is 6m.

**The greatest possible length of each plank is 6m.**

** **

**13.) A room measures 4.8m and 5.04m. Find the size of the largest square tile that can be used to tile the floor without cutting any tile.**

ANSWER:

Given that,

A room measures 4.8m and 5.04m.

We have to find the size of the largest square tile that can be used to tile the floor without cutting any tile.

We have to find HCF of 4.8m and 5.04m.

**HCF of 4.8m and 5.04m is 24 cm**

**The size of the largest square tile is 24 cm.**

**14.) Reduce each of the following fractions to lowest forms:-**

(i)85/102

ANSWER:

We have to reduce 85/102 to lowest forms.

85/102 we multiply and divide by 17

**85/102 = 17 x 5 / 17 x 6 = 5/6**

** **

(ii)52/130

ANSWER:

We have to reduce 52/130 to lowest forms.

52/130 we multiply and divide by 26

**52/130 = 26 x 2 / 26 x 5 = 2/5**

** **

(iii) 289/391

ANSWER:

We have to reduce 289/391 to lowest forms.

289/391 we multiply and divide by 17

**289/391 = 17 x 17 / 17 x 23 = 17/23**

** **

**Exercise 3.5**

**1.) Find LCM of following numbers by prime factorisation method:-**

(i) 45, 60

ANSWER:

We have to find LCM of 45, 60 by prime factorisation method.

We know,

The least common multiple of two or more given numbers is the lowest (or smallest or least) of their common multiples.

**LCM of 45, 60 = 5 x 3 x 3 x 4 = 180**

(ii) 52, 56

ANSWER:

We have to find LCM of 52, 56 by prime factorisation method.

We know,

The least common multiple of two or more given numbers is the lowest (or smallest or least) of their common multiples.

**LCM of 52, 56 = 4 x 13 x 14 = 728**

(iii) 96, 360

ANSWER:

We have to find LCM of 96, 360 by prime factorisation method.

We know,

The least common multiple of two or more given numbers is the lowest (or smallest or least) of their common multiples.

**LCM of 96, 360 = 2 x 2 x 2 x 3 x 4 x 15 = 1440**

** **

(iv) 36, 96, 180

ANSWER:

We have to find LCM of 36, 96, and 180 by prime factorisation method.

We know,

**LCM of 36, 96, and 180 = 2 x 2 x 3 x 3x 5 x 8 = 1440**

(v) 18, 42, 72

ANSWER:

We have to find LCM of 18, 42, and 72 by prime factorisation method.

We know,

LCM of 18, 42, and 72 = 2 x 3 x 3 x 4 x 7

**LCM of 18, 42, and 72 = 504**

** **

**2.) Find LCM of the following by common division method:-**

(i) 24, 64

ANSWER:

We have to find LCM of 24, 64 by common division method.

We know,

LCM of 24, 64 = 2 x 2 x 2 x 3 x 8

**LCM of 24, 64 = 192**

** **

(ii) 42, 63

ANSWER:

We have to find LCM of 42, 63 by common division method.

We know,

LCM of 42, 63 = 2 x 3 x 3 x 7

**LCM of 42, 63 = 126**

(iii) 108, 135, 162

ANSWER:

We have to find LCM of 108, 135, and 162 by common division method.

We know,

LCM of 108, 135, and 162 = 3 x 3 x 3 x 2 x 2 x 5 x 3

**LCM of 108, 135, and 162 = 1620**

** **

(iv) 16, 18, 48

ANSWER:

We have to find LCM of 16, 18, and 48 by common division method.

We know,

LCM of 16, 18, and 48 = 2 x 3 x 8 x 3

**LCM of 16, 18, and 48 = 144**

** **

(v) 48, 72, 108

ANSWER:

We have to find LCM of 48, 72, and 108 by common division method.

We know,

LCM of 48, 72, and 108 = 2 x 2 x 3 x 3 x 2 x 3 x 4

**LCM of 48, 72, and 108 = 432**

** **

**3.) Find the smallest number which is divisible by 6, 8 and 10.**

ANSWER:

We have to find the smallest number which is divisible by 6, 8 and 10.

We have to find LCM of 6, 8 and 10.

LCM of 6, 8 and 10 = 2 x 3 x 4 x 5

LCM of 6, 8 and 10 = 120

**The smallest number which is divisible by 6, 8 and 10 is 120.**

**4.) Find the least number when divided by 10, 12 and 15 leaves remainder 7 in each case.**

ANSWER:

We have to find the least number when divided by 10, 12 and 15.

We have to find LCM of 10, 12 and 15.

LCM of 10, 12 and 15 = 5 x 3 x 2 x 2

LCM of 10, 12 and 15 = 60

Leaves remainder 7 in each case

**60 + 7 = 67**

** **

**5.) Find the greatest 4-digit number exactly divisible by 12, 18 and 30.**

ANSWER:

We have to find the greatest 4-digit number exactly divisible by 12, 18 and 30.

LCM of 12, 18 and 30 = 2 x 2 x 3 x 3 x 5

LCM of 12, 18 and 30 = 180

The greatest 4-digit number = 180 x 55

**The greatest 4-digit number = 9900**

**6.) Find the smallest 4-digit number exactly divisible by 15, 24 and 36.**

ANSWER:

We have to find the smallest 4-digit number exactly divisible by 15, 24 and 36.

**The smallest 4-digit number exactly divisible by 15, 24 and 36 is 1080**

**7.) Four bells toll at intervals of 4, 7, 12 and 14 seconds. The bells toll together at 5a.m. When will they again toll together?**

ANSWER:

Given that,

Four bells toll at intervals of 4, 7, 12 and 14 seconds. The bells toll together at 5a.m.

We have to find when they will again toll together.

We have to find LCM of 4, 7, 12 and 14 seconds.

LCM of 4, 7, 12 and 14 seconds = 2 x 2 x 7 x 3

LCM of 4, 7, 12 and 14 seconds = 84 seconds

**They will again toll together at 5:01:24 a.m.**

** **

**8.) Three boys step off together from the same spot their steps measures 56cm, 70cm and 63cm respectively. At what distance from the starting point will they again step together?**

ANSWER:

Given that,

Three boys step off together from the same spot their steps measures 56cm, 70cm and 63cm respectively.

We have to find at what distance from the starting point they will again step together.

We have to find LCM of 56cm, 70cm and 63cm

LCM of 56cm, 70cm and 63cm = 7 x 2 x 4 x 5 x 9

LCM of 56cm, 70cm and 63cm = 2520 cm

**2520 cm distance from the starting point they will again step together.**

** **

**9.) Can two numbers have 15 as their HCF and 65 as their LCM. Give reasons in support of your answer.**

ANSWER:

We know,

HCF of given numbers is factor of their LCM.

But 15 is not factor of 65.

**There cannot 2 numbers have 15 as their HCF and 65 as their LCM**

**10.) Can two numbers have 12 as their HCF and 72 as their LCM. Give reasons in support of your answer.**

ANSWER:

We know,

HCF of given numbers is factor of their LCM.

12 is factor of 72.

**There can 2 numbers have 12 as their HCF and 72 as their LCM.**

**11.) The HCF and LCM of two numbers are 13 and 182 respectively. If one of the numbers is 26. Find other number.**

ANSWER:

Given that,

The HCF and LCM of two numbers are 13 and 182 respectively

one of the numbers is 26.

We have to find other number.

We know,

One number x other number = The HCF x LCM of two numbers

Other number = 13 x 182 / 26

**Other number = 91**

**Other number is 91.**

**12.) The LCM of two co-prime numbers is 195. If one number is 15 then find the other number.**

ANSWER:

Given that,

The LCM of two co-prime numbers is 195.

We know,

HCF of two co-prime numbers is 1.

One number is 15

We have to find other number.

We know,

One number x other number = The HCF x LCM of two numbers

Other number = 195 x 1 / 15

**Other number = 13**

** **

**13.) The HCF of two numbers is 6 and product of two numbers is 216. Find their LCM.**

ANSWER:

Given that,

The HCF of two numbers is 6 and product of two numbers is 216.

We have to find their LCM.

We know,

One number x other number = The HCF x LCM of two numbers

216 = 6 x LCM.

LCM = 216 / 6

**LCM = 36**

**Multiple Choice Questions**

**1.) Which number is a factor of every number?**

(a) 0

(b) 1

(c) 2

(d) 3

ANSWER: (b) 1

**2.) How many even numbers are prime?**

(a) 1

(b) 2

(c) 3

(d) 4

ANSWER: (a) 1

**3.) The smallest composite number is**

(a) 1

(b) 2

(c) 3

(d) 4

ANSWER: (d) 4

**4.) Which of the following number is a perfect number?**

(a) 8

(b) 6

(c) 12

(d) 18

ANSWER: (b) 6

**5.) Which of the following is not a multiple of 7?**

(a) 35

(b) 48

(c) 56

(d) 91

ANSWER: (b) 48

**6.) Which of the following is not a factor of 36?**

(a) 12

(b) 6

(c) 9

(d) 8

ANSWER: (d) 8

**7.) The number of prime numbers up to 25 are**

(a) 9

(b) 10

(c) 8

(d) 12

ANSWER: (a) 9

**8.) Which mathematician gave the method to find prime and composite numbers?**

(a) Aryabhatta

(b) Ramayan

(c) Eratosthenes

(d) Goldbach

ANSWER: (c) Eratosthenes

**9.) The statement “Every even number greater than 4 can be expressed as the sum of two odd prime numbers” is given by**

(a) Goldbach

(b) Eratosthenes

(c) Aryabhatta

(d) Ramanujan

ANSWER: (a) Goldbach

**10.) Which of the following is a prime number?**

(a) 221

(b) 195

(c) 97

(d) 111

ANSWER: (c) 97

**11.) Which of the following number is divisble by 4?**

(a) 52369

(b) 25746

(c) 21564

(d) 83426

ANSWER: (c) 21564

**12.) Which of the following is not true?**

(a) If a number is factor of two numbers then it is also factor of their sum

(b) If a number is factor of two numbers then it is also factor of their difference.

(c) 15 and 24 are co-prime to each other.

(d) 1 is neither prime nor composite.

ANSWER: (c) 15 and 24 are co-prime to each other.

**13.) Which of the following pair is co-prime?**

(a) (12, 25)

(b) (18, 27)

(c) (25, 35)

(d) (21, 56)

ANSWER: (a) (12, 25)

**14.) Which of the following number is divisible by 8?**

(a) 123568

(b) 412580

(c) 258124

(d) 453230

ANSWER: (a) 123568

**15.) Prime factorisation of 84**

(a) 2x2x3x2x7

(b) 7 x 2 x 3×3

(c) 2x3x7x2

(d) 3 x 2 x 3x2x7

ANSWER: (c) 2x3x7x2

**16.) HCF of 25 and 45 is**

(a) 15

(b) 5

(c) 225

(d) 135

ANSWER: (b) 5

**17.) If LCM of two numbers is 36 then which of the following cannot be their HCF?**

(a) 9

(b) 12

(c) 8

(d) 18

ANSWER: (c) 8

**18.) The LCM of two co-prime numbers is 143. If one number is 11 then find other number.**

(a) 132

(b) 154

(c) 18

(d) 13

ANSWER: (d) 13

**19.) Find the greatest number which divides 145 and 235 leaving the remainder 1 in each case.**

(a) 24

(b) 18

(c) 19

(d) 17

ANSWER: (b) 18

**20.) The greatest 4 digit number which is divisible by 12, 15 and 20**

(a) 9990

(b) 9000

(c) 9960

(d) 9999

ANSWER: (c) 9960

Do you have solved: Chapter 2 Whole Number Solution