RS Aggarwal Solutions: Factors & Multiples (Exercise 2A)

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) 75
Ans.
(i) We know that
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.
(ii) We know that
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
Ans.
(i) First five multiples of 17 are :

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) 253
Ans.
(i) 32 is a multiple of 2, so it is an even number.
(ii) 37 is not a multiple of 2, so it is an odd number.
(iii) 50 is a multiple of 2, so it is an even number.
(iv) 58 is a multiple of 2, so it is an even number.
(v) 69 is not a multiple of 2, so it is an odd number.
(vi) 144 is a multiple of 2, so it is an even number.
(vii) 321 is not a multiple of 2, so it is an odd number.
(viii) 253 is not a multiple of 2, so it is an odd number.

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 40
Sol.
(i) Prime numbers between 10 and 40 are :
11, 13, 17, 19, 23, 29, 31, 37.
(ii) Prime numbers between 80 and 100 are :
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.

Q.8. Find which of the following numbers are prime :

(i) 87
(ii) 89

(iii) 63
(iv) 91
Ans.

(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-primes
always prime ? If no, illustrate your answer by an example.
Ans.
Co-primes. Two numbers are said to  be co-prime if they do not have a common factor.

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
Ans. (i) 31 = 5 + 7 + 19
(ii) 35 = 5 + 7 + 23
(iii) 49 = 3 + 5 + 41
(iv) 63 = 7 + 13 + 43

Q.16. Express each of the following numbers as the sum of twin primes :

(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.