RS Aggarwal 2021-2022 for Class 6 Maths Chapter 2 - Factors and Multiples
EXERCISE 2A
1. Define: (i) factor (ii) multiple. Give five examples of each.
Solution:-
(i) Factor :- An exact divisor of that number is called Factor.
Example 1:- Factor of 15 is 1, 3, 5 and 15.
Example 2:- Factor of 18 is 1, 2, 3, 6, 9 and 18.
Example 3:- Factor of 19 is 1, and 19.
Example 4:- Factor of 20 is 1, 2, 4, 5, 10 and 20.
Example 5:- Factor of 24 is 1, 2, 3, 4, 6, 8, 12 and 24.
(ii) Multiple:- A number obtained by multiplying it by a natural number is called Multiple.
Example 1:- 18 is Multiple of 1, 2, 3, 6, 9 and 18.
Example 2:- 15 is Multiple of 1, 3, 5, and 15.
Example 3:- 19 is Multiple of 1, and 19.
Example 4:- 20 is Multiple of 1, 2, 4, 5, 10 and 20.
Example 5:- 24 is Multiple of 1, 2, 3, 4, 6, 8, 12 and 24.
2. Write down all the factors of
(i) 20 (ii) 36 (iii) 60 (iv) 75
Solution:-
(i) 20 = 1 x 20; 20 = 2 x 10; 20 = 4 x 5
(20 comes in the table of 1, 2, 4, 5, 10 and 20)
So, The factors of 20 are 1, 2, 4, 5, 10 and 20.
(ii) 36 = 1 × 36; 36 = 2 × 18; 36 = 3 × 12 and 36 = 4 × 9
(36 comes in the table of 1, 2, 3, 4, 6, 9, 12 and 36)
So, The factors of 36 are 1, 2, 3, 4, 6, 9, 12 and 36.
(iii) 60 = 1 × 60; 60 = 2 × 30; 60 = 3 × 20; 60 = 4 × 15 and 60 = 5 × 12
(60 comes in the table of 1, 2, 3, 4, 5, 6, 10, 12, 15 and 60)
So, The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15 and 60.
(iv) 75 = 1 × 75; 75 = 3 × 25 and 75 = 5 × 15
(75 comes in the table of 1, 3, 5, 15, 25 and 75)
So, The factors of 75 are 1, 3, 5, 15, 25 and 75.
(i) 17 (ii) 23 (iii) 65 (iv) 70
Solution:-
(i) 17 × 1 = 17; 17 × 2 = 34; 17 × 3 = 51; 17 × 4 = 68 and 17 × 5 = 85
∴ The first five multiples of 17 are 17, 34, 51, 68 and 85.
(ii) 23 × 1=23; 23 × 2 = 46; 23 × 3 = 69; 23 × 4 = 92 and 23 × 5 = 115
∴ The first five multiples of 23 are 23, 46, 69, 92 and 115.
(iii) 65 × 1 = 65; 65 × 2 = 130; 65 × 3 = 195; 65 × 4 = 260 and 65 × 5 = 325
∴ The first five multiples of 65 are 65, 130, 195, 260 and 325.
(iv) 70 × 1=70; 70 × 2 = 140; 70 × 3 = 210; 70 × 4 = 280 and 70 × 5 = 350
∴ The first five multiples of 70 are 70, 140, 210, 280 and 350.
Note:- Table of any number is multiple of that number.
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
Solution:-
(i) 32 (iii) 50 (iv) 58 (vi) 144 are Even Number (Even number ending with 0,2,4,6 and 8)
(ii) 37 (v) 69 (vii) 321 (viii) 253 are Odd Number (Odd number ending with 1,3,5,7 and 9)
5. What are prime numbers? Give ten examples.
Solution:-
Prime Number :- A number which has only two factors, namely 1 and itself is called Prime Number.
Or, A number which is divisible by 1 and itself is called Prime Number.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 are prime numbers.
6. Write all the prime numbers between
(i) 10 and 40 (ii) 80 and 100 (iii) 40 and 80 (iv) 30 and 40
Solution:-
(i) All prime numbers between 10 and 40 are 11, 13, 17, 19, 23, 29, 31 and 37.
(ii) All prime numbers between 80 and 100 are 83, 89 and 97.
(iii) All prime numbers between 40 and 80 are 41, 43, 47, 53, 59, 61, 67, 71, 73 and 79.
(iv) All prime numbers between 30 and 40 are 31 and 37.
(ii) List all even prime numbers.
(iii) Write the smallest odd prime number.
Solution:-
(i) The smallest prime number is 2.
(ii) There is only one even prime number, i.e., 2.
(iii) The smallest odd prime number is 3.
8. Find which of the following numbers are primes:
(i) 87 (ii) 89 (iii) 63 (iv) 91
Solution:-
(i) Factor of 87 are 1, 3, 29 and 87.
87 has more than 2 factors. Therefore 87 is not a prime number.
(ii) Factor of 89 are 1 and 89.
89 has only two factor. Therefore 89 is a prime number.
(iii) Factor of 63 are 1, 3, 7, 9, 21 and 63.
63 has more than 2 factors. Therefore 63 is not a prime number.
(iv) Factor of 91 are 1, 7, 13 and 91.
91 has more than 2 factors. Therefore 91 is not a prime number.
Hint. See the sieve of Eratosthenes.
Solution:-
90, 91, 92, 93, 94, 95 and 96 are seven consecutive numbers and none of them is a prime.
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.
Solution:-
(i) No, there are no counting numbers with no factors at all because every number has at least two factors, i.e., 1 and itself.
(ii) There is only one number that has exactly one factor, i.e, 1.
(iii) The numbers between 1 and 100 that have exactly three factors are 4, 9, 25 and 49.
Solution:-
The numbers that have more than two factors are known as composite numbers.
Yes, a composite number can be odd.
The smallest odd composite number is 9.
Solution:-
Two consecutive odd prime numbers are called twin primes.
Or, Two prime number which difference is 2 is called Twin Prime number.
The pairs of twin primes between 50 to 100 are (59, 61) and (71, 73).
Solution:-
Two numbers which have only one common factor as 1, are said to be co-primes.
Five pairs of co primes: (i) 2 and 3 (ii) 3 and 4 (iii) 4 and 5 (iv) 4 and 9 (v) 8 and 15
No, co–primes are not always primes.
For example, 7 and 9 are co-prime numbers, where 7 is a prime number and 9 is not a prime number.
14. Express each of the following numbers as the sum of two odd primes:
(i) 36 (ii) 42 (iii) 84 (iv) 98
Solution:-
(i) The sum of two odd prime numbers of 36 is (36 = 29 + 7).
(ii) The sum of two odd prime numbers of 42 is (42 = 29 + 13).
(iii) The sum of two odd prime numbers of 84 is (84 = 41 + 43).
(iv) The sum of two odd prime numbers of 98 is (98 = 31 + 67).
(i) 31 (ii) 35 (iii) 49 (iv) 63
Solution:-
(i) The sum of three odd prime numbers of 31 as (31 = 5 + 7 + 19).
(ii) ) The sum of three odd prime numbers of 35 as (35 = 17 + 13 + 5).
(iii) The sum of three odd prime numbers of 49 as (49 = 13 + 17 + 19).
(iv) The sum of three odd prime numbers of 63 as (63 = 29 + 31 + 3).
(i) 36 (ii) 84 (iii) 120 (iv) 144
Solution:-
(i) 36 can be expressed as the sum of twin primes as (36 = 17 + 19).
(ii) 84 can be expressed as the sum of twin primes as (84 = 41 + 43).
(iii) 120 can be expressed as the sum of twin primes as (120 = 59 + 61).
(iv) 144 can be expressed as the sum of twin primes as (144 = 71 + 73).
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.
(ii) False. 2 is an even prime number.
(iii) False. 3 and 7 are two prime numbers and their sum is 10, which is even.
(iv) False. 4 and 9 are co-primes but neither of them is a prime number.
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