**Playing with Numbers**

**Question 1.****Find the common factors of:(a)** 20 and 28

**(b)**15 and 25

**(c)**35 and 50

**(d)**56 and 120

**Solution:****(a)** We have, 20 = 1 x 20

= 2 x 10

= 4 x 5

∴ All the factors of 20 are 1, 2, 4, 5, 10 and 20

Again, 28 = 1 x 28

28 = 2 x 14

28 = 4 x 7

∴ All the factors of 28 are 1, 2, 4, 7, 14 and 28.

Out of these 1, 2 and 4 occur in both the lists.

∴ 1, 2 and 4 are common factors of 20 and 28.

**(b)** We have, 15 = 1 x 15

15 = 3 x 5

∴ All the factors of 15 are 1, 3, 5 and 15.

Again, 25 = 1 x 25

25 = 5 x 5

∴ All the factors of 25 are 1, 5 and 25.

Out of these 1 and 5 occur in both the lists.

∴1 and 5 are common factors of 15 and 25.

**(c)** We have, 35 = 1 x 35

35 = 5 x 7

∴ All the factors of 35 are 1, 5, 7 and 35.

Again, 50 = 1 x 50

50 = 2 x 25

50 = 5 x 10

∴ All the factors of 50 are 1, 2, 5, 10, 25 and 50. ,

Out of these 1 and 5 occur in both the lists.

∴ 1 and 5 are common factors of 35 and 50.

**(d)** We have, 56 = 1 x 56

56 = 2 x 28

56 = 4 x 14

56 = 7 x 8

∴ All the factors of 56 are 1, 2, 4, 7, 8, 14, 28 and 56.

Again, 120 = 1 x 120

120 = 2 x 60

120 = 3 x 40

120 = 4 x 30

120 = 5 x 24

120 = 6 x 20

120 = 8 x 15

120 = 10 x 12

∴ All the factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.

Out of these 1, 2, 4 and 8 occur in both the lists.

∴ 1, 2, 4 and 8 are common factors of 56 and 120.

**Question 2.**

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

**Solution:****(a)** We have, 4=1 x 4

4 = 2 x 2

∴ All the factors of 4 are 1, 2 and 4.

Again, 8 = 1 x 8

8 = 2 x 4

∴All the factors of 8 are 1, 2, 4 and 8.

Again, 12 = 1 x 12

12 = 2 x 6

12 = 3 x 4

∴ All the factors of 12 are 1, 2, 3, 4, 6 and 12.

Out of these 1, 2 and 4 occur in all the three lists.

∴ 1, 2 and 4 are common factors of 4, 8 and 12.

**(b)** We have, 5 = 1 x 5

∴All the factors of 5 are 1 and 5.

15 = 1 x 15

15 = 3 x 5

∴ All the factors of 15 are 1, 3, 5 and 15.

25 = 1 x 25

25 = 5 x 5

∴ All the factors of 25 are 1, 5 and 25.

Out of these 1 and 5 occur in all the three lists.

∴ 1 and 5 are common factors of 5, 15 and 25.

**Question 3.****Find first three common multiples of:(a)** 6 and 8

**(b)**12 and 18

**Solution:****(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,…

Out of these 24, 48, 72, … occur in both thfe lists.

∴ The first three common multiples of 6 and’8 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, …

Out of these 36, 72, 108, … occur in both the lists.

∴ The first three common multiples of 12 and 18 are 36, 72 and 108.

**Question 4.****Write all the numbers less than 100 which are corrimon multiples of 3 and 4.**

**Solution:**

Common multiples of 3 and 4 are multiples of 3 x 4 i. e., 12.

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

**Question 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

**Solution:****(a)** Factors of 18 are 1, 2, 3, 6, 9 and 18 and, that of 35 are 1, 5, 7 and 35.

∴ Common factor of 18 and 35 is 1.

Thus, 18 and 35 are co-prime.

**(b)** Factors of 15 are 1, 3, 5 and 15 and, that of 37 are 1 and 37.

∴ Common factor of 15 and 37 is 1.

Thus, 15 and 37 are co-prime.

**(c)** Since 5 is a common factor of 30 and 415.

∴ 30 and 415 are not co-prime.

**(d)** ∴ 68 + 17 = 4 i.e., 17 is a common factor of 17 and 68.

∴ 17 and 68 are not co-prime.

**(e)** Since 1 is the only common factor of 216 and 215.

∴ 216 and 215 are co-prime.

**(f)** Since 1 is the only common factor of 81 and 16.

∴ 81 and 16 are co-prime.

**Question 6.**

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

**Solution:**

Since a number is divisible by both 5 and 12.

So, it is also divisible by 5 x 12 i. e., 60.

**Question 7.**

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

**Solution:**

Factors of 12 are 1, 2, 3, 4 and 12.

Since a number is divisible by 12. So, it is also divisible by the factors of 12.

Thus, the number is also divisible by 2, 3 and 4.