** **

** Playing with numbers Exercise 3.1**

**Q 1. 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 × 6 24 = 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

**(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

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

**Q. 2 Write first five multiplies 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.

Q. 3 **Match the items in column 1 with the items in column 2.**

**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) 25 (e) Factor of 50**

** (f) Factor 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 (f) Factor of 20

(v) 25 (e) Factor of 50

**Q.4 Find all the multiples of 9 up to 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

** Playing with numbers Exercise 3.2**

**Q. 1 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

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

e.g., 2 + 4 = 6

10 + 18 = 28

**Q. 2 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 × 5 × 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 = 5, i.e., odd

(h) True

(i) False 2 is a prime number

(j) True 2 × 4 = 8, i.e., even

**Q. 3 The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.**

**Answer :**

17, 71

37, 73

79, 97

**Q. 4 Write down separately the prime and composite numbers less than 20.**

**Answer :**

Prime numbers less than 20 are

2, 3, 5, 7, 11, 13, 17, 19

Composite numbers less than 20 are

4, 6, 8, 9, 10, 12, 14, 15, 16, 18

**Q. 5 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.

**Q. 6 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

**Q. 7 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

**Q. 8 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.

(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.

**Q. 9 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, 94

95 1, 5, 19, 95

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

**Q. 10 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

**Q. 11 Write five pairs of prime numbers less than 20 whose sum is divisible by 5.**

**(Hint: 3 + 7 = 10)**

**Answer:**

2 + 3 = 5

2 + 13 = 15

3 + 17 = 20

7 + 13 = 20

19 + 11 = 30

**Q. 12 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

** **

** Playing with numbers Exercise 3.3**

Q. 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):**

**Answer:**

**Q. 2: 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**

**(b) 726352**

**(c) 5500**

**(d) 6000**

**(e) 12159**

**(f) 14560**

**(g) 21084**

**(h) 31795072**

**(i) 1700**

**j) 2150**

**Q.3: 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**

**Answer:**

**(a) 297144**

**(b) 1258**

**(c) 4335**

**d) 61233**

**(e) 901352**

**(f) 438750**

**(g) 1790184**

**(h) 12583**

**(i) 639210**

**(j) 17852**

**Q 4: 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**

**(b) 10824**

**(c) 7138965**

**(d) 70169308**

**(e) 10000001**

**(f) 901153**

**Q. 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**

**(b) 4765 ___2**

**Answer**:

**(a) _6724**

**(b) 4765_2**

**. 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**

**(b) 8_9484**

**Playing with numbers Exercise 3.4**

**Q. 1: Find the common factors of:**

**(a) 20 and 28**

**(b) 15 and 25**

**(c) 35 and 50**

**(d) 56 and 120**

**Answer :**

**Q. 2: Find the common factors of:**

**(a) 4, 8 and 12**

**(b) 5, 15 and 25**

**Answer :**

**(a) 4, 8, 12**

**(b) 5, 15, and 25**

**Q. 3: Find first three common multiples of:**

**(a) 6 and 8**

**(b) 12 and 18**

**Answer :**

**(a) 6 and**

**8**

**(b) 12 and 18**

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

**Answer**:

**Q. 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**

**Answer**:

**(a) 18 and 35**

**b) 15 and 37**

**(c) 30 and 415**

**(d) 17 and 68**

**(e) 216 and 215**

**(f) 81 and 16**

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

**Answer :**

**Q.7: A number is divisible by 12. By what other number will that number be divisible**?

**Answer :**

**Playing with numbers Exercise 3.5**

**Q. 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.**

**Answer :**

**False**

**True**, as 9 = 3 × 3

**False**

**True**, as 9 × 10 = 90

**False**

**False**

**True**, as 8 = 2 × 4

**True**

**False**

**Answer : **