# Class 6 Mathematics Chapter 3 Solutions | Playing With Numbers | Exercise 3.4 and 3.5

Class 6 Mathematics Chapter 3 | Exercise 3.4

1. Find the common factors of:

a) 20 and 28

b) 15 and 25

c) 35 and 50

d) 56 and 120

Ans.

a) Factors of 20 are 1, 2,4,5,10,20

Factors of 28 are 1, 2,4,7,14,28.

Common factors are 1, 2, and 4

b) Factors of 15 are 1, 3,5,15

Factors of 25 are 1,5,25

Common factors are 1 and 5

c) Factors of 35 are 1, 5,7,35

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

Common factors are 1 and 5

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

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

Common factors are 1,2,4,8

2. Find the common factors of:

a) 4, 8 and 12

b) 5, 15 and 25

Ans.

a) Factors of 4 are 1, 2, 4

Factors of 8 are 1,2,4,8

Common factors are 1, 2, 4

b) Factors of 5 are 1 and 5

Factors of 25 are 1,5,25

Common factors are 1 and 5

3. Find first three common multiples of:

a) 6 and 8

b) 12 and 18

Ans.

a) Multiples of 6 are 6,12,18,24,30,36,42,48,54,60,66,72…………….

Multiples of 8 are 8,16,24,32,40,48,56,64,72,80……..

Common multiples are 24, 48, and 72

b) Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96,108……….

Multiples of 18 are 18, 36, 54, 72, 90,108………

Common multiples are 36, 72 and 108

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

Ans. Multiples of 3 are 3,6,9, 12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99

Multiples of 4 are 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100

Common multiples of 3 and 4 less than 100 are 12, 24,36,48,60,72,84,96

5. 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

Ans.

a) 18 and 35

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

Factors of 35 are 1, 5,7,35

Common factor = 1

Since 18 and 35 have only one as the common factor, 18 and 35 are co-prime numbers

b) 15 and 37

Factors of 15 are 1, 3, 5 and 15

Factors of 37 are 1 and 37

Common factor = 1

So 15 and 37 are co-prime.

c) 30 and 415

Factors of 30 are 1, 2, 3, 5, 6, 10, 30

Factors of 415 are 1,5,83,415

Common factors are 1 and 5

So these numbers are not co-prime number.

d) 17 and 68

Factors of 17 are 1 and 17

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

Common factors are 1 and 17

So these numbers are not co prime numbers.

e) 216 and 215

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

Factors of 215 are 1,5,43,215

Common factor = 1

Therefore the given numbers are co prime.

f) 81 and 16

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

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

Common factor = 1

Therefore the given numbers are co prime

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

Ans.

Factors of 5 are 1 and 5

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

Common factor = 1

So these numbers are co prime and the number will also be divisible by their product, which is 60.

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

Ans.

Since the number is divisible by 12,it will also be divisible by its factors .

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

So 1,2,3,4 and 6 are numbers other than 12 by which this number is also divisible.

### EXERCISE 3.5

1. 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.

Ans.

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

b) True.

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

d) True

e) False, 16 and 81 are co prime but both are composite numbers.

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

g) True

h) True

i) False, 2 is divisible by 10 but cannot be divided by 7 and 3.

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

**Answers**

3. Which factors are not included in the prime factorization of a composite number?

Ans. One and the number itself.

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

Ans. 9999 is the greatest 4-digit number.

9999 can be expressed as 3 x 3 x 11 x 101

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

Ans. 10000 is the smallest 5 digit number.

10000 can be expressed as 2 x 2 x 2 x2 x5 x 5 x5 x 5

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

Ans. 1729 can be expressed as 7 x 13 x 19

The difference of two consecutive prime numbers is 6.

i.e. 13 – 7 = 6

19 – 13= 6

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

Ans. i) 2 x 3 x 4 = 24 is divisible by 6.

ii) 5 x 6 x 7 = 210 is divisible by 6.

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

Ans. 3 +5 =8 which is divisible by 4.

19 + 21 = 40 which is divisible by 4.

9. In which of the following expressions, prime factorization has been done?

a) 24 = 2 x 3 x 4

b) 56 = 7 x 2 x 2 x 2

c) 70 = 2 x 5 x 7

d) 54 = 2 x 3 x 9

Ans.

a) Since 4 is composite, prime factorization has not been done.

b) Since all factors are prime, prime factorization has been done.

c) Since all factors are prime, prime factorization has been done.

d) Since 9 is composite, prime factorization has not been done.

10. 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)

Ans.

45 = 5 x 9

Factors of 5 are 1 and 5

Factors of 9 are 1, 3, 9

Common factor = 1

So 5 and 9 are co – prime numbers.

Since the last digit of 25110 is 0,it is divisible by 5.

Sum of the digits = 2 + 5 + 1 + 1 + 0 = 9, which is divisible by 9.

So 25110 is divisible by 9.

Since 25110 is divisible by both 5 and 9, 25110 is divisible by 45 also.

11. 18 is divisible by both 2 and 3.It is also divisible by 2 x 3 =6.Similarly, a number is divisible by both 4 and 6.Can we say that the number must also be divisible by 4 x 6 =24? If not, give an example to justify your answer.

Ans. No, Number 12 is divisible by both 4 and 6.

But 12 is not divisible by 24.

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

Ans. 210 = 2 x 3 x 5 x7