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**Q.1. Define (i) factor (ii) multiple. Give five**

**examples of each.****Ans.****(i) **A factor of a number is an exact

divisor of that number.**Examples :****1. **2 is a factor of 8**2.** 5 is a factor of 15**3. **9 is a factor of 27**4. **4 is a factor of 20**5. **3 is a factor of 12.

**(ii) Multiple. **A number is said to be a

multiple of any of its factors.

**Examples :****1. **15 is a multiple of 3

**2.** 8 is a multiple of 4

**3.** 10 is a multiple of 2

**4. **25 is a multiple of 5

**5.** 18 is a multiple of 9.**Q.2. Write down all the factors of :(i) 20(ii) 36(iii) 60(iv) 75Ans**

20 = 1 × 20, 20 = 2 × 10, 20 = 4 × 5

which shows that the numbers 1, 2, 4,

5, 10, 20 exactly divide 20.

∴ 1, 2, 4, 5, 10 and 20 are all factors of 20.

36 = 1 × 36, 36 = 2 × 18, 36 = 3 × 12,

36 = 4 × 9, 36 = 6 × 6

This shows that each of the numbers 1,

2, 3, 4, 6, 9, 12, 18, 36 exactly divides 36.

∴ 1, 2, 3, 4, 6, 9, 12, 18, 36 are the factors of 36.**(iii)** We know that

60 = 1 × 60, 60 = 2 × 30, 60 = 3 × 20,

60 = 4 × 15, 60 = 5 × 12, 60 = 6 × 10

This shows that each of the numbers 1,2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

exactly divides 60.

∴ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 are all the factors of 60.**(iv)** We know that

75 = 1 × 75, 75 = 3 × 25, 75 = 5 × 15

This shows that each of the numbers 1,3, 5, 15, 25, 75 exactly divides 75.

∴ 1, 3, 5, 15, 25, 75 are all the factors of 75.**Q.3. Write the first five multiples of each of the following numbers :**

**(i) 17(ii) 23(iii) 65(iv) 70**

17 × 1 = 17

17 × 2 = 34

17 × 3 = 51

17 × 4 = 68

17 × 5 = 85**(ii)** First five multiples of 23 are :

23 × 1 = 23

23 × 2 = 46

23 × 3 = 69

23 × 4 = 92

23 × 5 = 115

**(iii)** First five multiples of 65 are :

65 × 1 = 65

65 × 2 = 130

65 × 3 = 195

65 × 4 = 260

65 × 5 = 325**(iv)** First five multiples of 70 are :

70 × 1 = 70

70 × 2 = 140

70 × 3 = 210

70 × 4 = 280

70 × 5 = 350**Q.4. Which of the following numbers are even and which are odd.**

**(i) 32(ii) 37**

**(iii) 50(iv) 58**

**(v) 69(vi) 144**

**(vii) 321**

**(viii) 253Ans.**

Q.5. What are prime numbers ? Give ten examples.

**Ans.****Prime Numbers. **Each of the numbers which has exactly two factors, namely 1 and itself, is called a prime number.**Examples.** The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are all prime numbers.**Q.6. Write all the prime numbers between :(i) 10 and 40(ii) 80 and 100**

**(iii) 40 and 80**

**(iv) 30 and 40Sol.**

11, 13, 17, 19, 23, 29, 31, 37.

83, 89, 97.

**(iii)** Prime numbers between 40 and 80 are :

41, 43, 47, 53, 59, 61, 67, 71, 73, 79**(iv) **Prime numbers between 30 and 40 are :

31, 37.**Q. 7. (i) Write the smallest prime number.(ii) List all even prime numbers.**

**(iii) Write the smallest odd prime number.Ans. **(i) 2 is the smallest prime number.

(ii) 2 is the only even prime number.

(iii) 3 is the smallest odd prime number.

**(i) 87(ii) 89**

**(iii) 63(iv) 91**

(i) We know that

87 = 1 × 87, 87 = 3 × 29

This shows that 1, 3, 29, 87 are the factors of 87.

∴ The number 87 is not a prime number

as it has more than 2 factors.

**(ii)** We have 89 = 1 × 89

∴ The number 89 is a prime number as

it has only 2 factors.

**(iii) **We have 63 = 1 × 63, 63 = 3 × 21,

63 = 7 × 9

This shows that the number 63 has more than 2 factors namely 1, 3, 7, 9, 21, 63.

So, it is not a prime number.

**(iv) **We have 91 = 1 × 91, 91 = 7 × 13

This shows that the number 91 has more than 2 factors namely 1, 7, 13, 91. So, it is not a prime number.**Q.9. Make a list of seven consecutive numbers, none of which is prime.**

**Ans. **From the Sieve of Eratosthenes, we see that the seven consecutive numbers are

90, 91, 92, 93, 94, 95 and 96.**Q.10. (i) Is there any counting number having no factor at all ?**

**(ii) Find all the numbers having exactly one factor.**

**(iii) Find numbers between 1 and 100 having exactly three factors.**

**Ans.**** (i) **There is no counting number having no factor at all.**(ii)** The number 1 has exactly one factor.**(iii)** The numbers between 1 and 100 having exactly three factors are : 4, 9, 25, 49.**Q.11. What are composite numbers ? Can a composite number be odd ? If yes, write**

**the smallest odd composite number.****Ans. ****Composite Numbers.** Numbers having more than two factors are called composite numbers. A composite number can be an odd number. The smallest odd composite number is 9.**Q.12. What are twin primes ? Write all the pairs**

**of twin primes between 50 and 100.**

**Ans.** **Twin-primes. **Two consecutive odd prime numbers are known as twinprimes.

The prime numbers between 50 and 100 are :

53, 59, 61, 67, 71, 73, 79, 83, 89, 97

From above pairs of twin-primes are

(59, 61), (71, 73)**Q.13. What are co-primes ? Give examples of five pairs of co-primes. Are co-primesalways prime ? If no, illustrate your answer by an example.**

**Examples.** Five pairs of co-primes are:

(i) 2, 3

(ii) 3, 4

(iii) 4, 5

(iv) 8, 15

(v) 9, 16

Co-primes are not always prime.

**Illustration.** In the pair (3, 4) of coprimes, 3 is a prime number whereas 4 is a composite number.**Q.14. Express each of the following numbers**

**as the sum of two odd primes :**

**(i) 36(ii) 42**

**(iii) 84(iv) 98**

**Sol.**

(i) 36 = 7 + 29**(ii)** 42 = 5 + 37**(iii)** 84 = 17 + 67**(iv) **98 = 19 + 79**Q.15. Express each of the following odd numbers as the sum of three odd prime numbers :(i) 31(ii) 35**

**(iii) 49(iv) 63**

**(i) 36(ii) 84**

**(iii) 120(iv) 144**

**Ans.****(i) **36 = 17 + 19**(ii) **84 = 41 + 43**(iii) **120 = 59 + 61**(iv)** 144 = 71 + 73**Q.****17. Which of the following statements are true ?(i) 1 is the smallest prime number.(ii) If a number is prime, it must be odd.(iii) The sum of two prime numbers is always a prime number.**

**(iv) If two numbers are co-prime, at least one of them must be a prime number.****Ans.**

(i) to (iv). None of the given statements

is true.

**TESTS FOR DIVISIBILITY OF NUMBERS**

**(i) Test of divisibility by 10.** A number is divisible by 10, if its unit’s digit is zero.

**(ii) Test of divisibility by 5.** A number is divisible by 5, if its unit’s digit is 0 or 5.

**(iii) Test of divisibility by 2. **A number is divisible by 2, if its unit’s digit is 0, 2, 4, 6 or 8.

**(iv) Test of divisibility by 3.** A number is divisible by 3, if the sum of its digits is divisible by 3.

**(v) Test of divisibility by 9.** A number is divisible by 9, if the sum of its digits is divisible by 9.**(vi) Test of divisibility by 4.** A number is divisible by 4, if the number formed by its digits in ten’s and unit’s places is divisible by 4.**GENERAL PROPERTIES OF DIVISIBILITY**

**Property 1. **If a number is divisible byanother number, it must be divisible by each of the factors of that number.**Property 2.** If a number is divisible by each of the two co-prime numbers, it must be divisible by their product.**Property 3.** If a number is a factor of each of the two given numbers then it must be a factor of their sum.

**Property 4.** If a number is a factor of each of the two given numbers then it must be a factor of their difference.

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