## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 4 Playing with Numbers Ex 4.1

Question 1.
Fill in the blanks:
(i) A number having exactly two factors is called a …..
(ii) A number having more than two factors is called a ………
(iii) 1 is neither ……… nor ………
(iv) The smallest prime number is ………
(v) The smallest odd prime number is ………
(vi) The smallest composite number is ………
(vii) The smallest odd composite number is ………
(viii)All prime numbers (except 2) are ………
Solution:
(i) A number having exactly two factors is called a prime number.
(ii) A number having more than two factors is called a composite number.
(iii) 1 is neither prime nor composite.
(iv) The smallest prime number is 2.
(v) The smallest odd prime number is 3.
(vi) The smallest composite number is 4.
(vii) The smallest odd composite number is 9.
(viii) All prime numbers (except 2) are odd numbers.

Question 2.
State whether the following statements are ture (T) or false (F):
(i) The sum of three odd numbers is an even number.
(ii) The sum of two odd numbers and one even number is an even number.
(iii) The product of two even numbers is always an even number.
(iv) The product of three odd numbers is an odd number.
(v) If an even number is divided by 2, the quotient is always an odd number.
(vi) All prime numbers are odd.
(vii) All even numbers are composite.
(viii) Prime numbers do not have any factors.
(ix) A natural number is called a composite number if it has atleast one more factor other than 1 and the number itself.
(x) Two consecutive numbers cannot be both prime.
(xi) Two prime numbers are always co-prime numbers.
Solution:
(i) False
(ii) True
(iii) True
(iv) True
(v) False
(vi) False
(vii) False
(viii) False
(ix) True
(x) False
(xi) True

Question 3.
Write all the factors of the following natural numbers:
(i) 68
(ii) 27
(iii) 210
Solution:
(i) 68
The factors of 68 are : 1, 2, 4, 17, 34, 68
(ii) 27
The factors of 27 are : 1, 3, 9, 27
(iii) 210
The factors of 210 are :
1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210

Question 4.
Write first six multiples of the following natural numbers:
(i) 3
(ii) 5
(iii) 12
Solution:
(i) 3
The first six multiples of 3 are
3, 6, 9, 12, 15, 18
(ii) 5
The first six multiples of 5 are
5, 10, 15, 20, 25, 30
(iii) 12
The first six multiples of 12 are
12, 24, 36, 48, 60, 72

Question 5.
Match the items in column 1 with the items in column 2:

 Column 1 Column 2 (i) 15 (a) Multiple of 8 (ii) 36 (b) Factor of 30 (iii) 16 (c) Multiple of 70 (iv) 20 (d) Factor of 50 (v) 25 (e) Multiple of 9 (vi) 210 (f) Factor of 20

Solution:

 Column 1 Column 2 (i) 15 (b) Factor of 30 (ii) 36 (e) Multiple of 9 (iii) 16 (a) Multiple of 8 (iv) 20 (f) Factor of 20 (v) 25 (d) Factor of 50 (vi) 210 (c) Multiple of 70

Question 6.
Find the common factors of :
(i) 20 and 28
(ii) 35 and 50
(iii) 56 and 120
Solution:
(i) 20 and 28
The factors of 20 are:
1, 2, 4, 5, 10, 20 The factors of 28 are:
1,2,4,7,14,28
The common factors of 20 and 28 are: 1, 2, 4

(ii) 35 and 20
The factors of 35 are:
1, 5, 7, 35
The factors of 20 are:
1, 2, 4, 5, 10, 20
The common factors of 35 and 20 are 1,5

(iii) 56 and 120
The factors of 56 are:
1, 2, 4, 7, 8, 14, 28, 56
The factors of 120 are:
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60,120
The common factors of 56 and 120 are 1, 2, 4,8

Question 7.
Find the common factors of:
(i) 4, 8, 12
(ii) 10, 30 and 45
Solution:
(i) The factors of 4 are:
1, 2, 4
The factors of 8 are:
1, 2, 4, 8
The factors of 12 are:
1,2, 3,4, 6, 12
The common factors of 4, 8, 12 are 1, 2, 4

(ii) 10, 30 and 45 .
The factor of 10 are:
1, 2, 5, 10
The factor of 30 are:
1, 2, 3, 5, 10, 15, 30
The factor of 45 are:
1, 3, 5, 9, 15, 45
The common factors of 10, 30, 45 are 1, 5

Question 8.
Write all natural numbers less than 100 which are common multiples of 3 and 4.
Solution:
Multiples of 3 are : 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87,
90, 93, 96, 99, 102, 105, 108, ……….
Multiples of 4 are : 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, ……………
∴ Common multiples of 3 and 4 are : 12, 24, 36, 48, 60, 72, 84, 96, 108, ………..
All the numbers less than 100 which are common multiples of 3 and 4 are 12, 24, 36, 48, 60, 72, 84 and 96.

Question 9.
(i) Write the odd numbers between 36 and 53.
(ii) Write the even numbers between 232 and 251.
Solution:
(i) The odd numbers between 36 and 53 are:
37, 39, 41, 43, 45, 47, 49, 51.
(ii) The even numbers between 232 and 251 are: 234, 236, 238, 240, 242, 244, 246, 248, 250.

Question 10.
(i) Write four consecutive odd numbers succeeding 79.
(ii) Write three consecutive even numbers preceding 124.
Solution:
(i) Four consecutive odd numbers succeeding 79 are : 81, 83, 85, 87.
(ii) Three consecutive even numbers preceding 124 are : 118, 120, 122.

Question 11.
What is greatest prime number between 1 and 15?
Solution:
The greatest prime number between 1 and 15 is 13.

Question 12.
Which of the following numbers are prime?
(i) 29
(ii) 57
(iii) 43
(iv) 61
Solution:
(i) 29
We have, 29 = 1 × 29
⇒ 29 has exactly two factors 1 and 29 itself.
∴ 29 is a prime number.

(ii) 57
We have, 57 = 1 × 57 = 3 × 19 = 57
∴ Factors of 57 are 1, 3, 19 and 57
⇒ 57 has more than two factors
∴ 57 is not a prime.

(iii) 43
We have, 43 = 1 × 43
⇒ 43 has exactly two factors 1 and 43 itself.
∴ 43 is a prime number.

(iv) 61
We have, 61 = 1 × 61
⇒ 61 has exactly two factors 1 and 61 itself.
∴ 61 is a prime number.

Question 13.
Which of the following pairs of numbers are co-prime?
(i) 12 and 35
(ii) 15 and 37
(iii) 27 and 32
(iv) 17 and 85
(v) 515 and 516
(vi) 215 and 415
Solution:
(i) 12 and 35
The factors of 12 are 1,2, 3,4, 6, 12
The factors of 35 are 1, 5, 7, 35
Since, the common factor of 12 and 35 is 1
∴ They are co-prime.

(ii) 15 and 37
The factors of 15 are 1, 3, 5, 15
The factors of 37 are 1, 37
The common factor of 15 and 37 is 1
∴ They are co-prime.

(iii) 27 and 32
The factors of 27 are 1, 3, 9, 27
The factors of 32 are 1, 2, 4, 8, 16, 32
Since, the common factor of 27 and 32 is 1 They are co-prime

(iv) 17 and 85
The factors of 17 are 1, 17
The factors of 85 are 1, 5, 17, 85
The common factors of 17 and 85 are 1 and 17
∴ They are not co-prime because they have more than 1 common factor.

(v) 515 and 516
The factors of 515 are 1, 5, 103, 515
The factors of 516 are 1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516
Since, the common factor of 515 and 516 are 1 and 5
∴ So, they are not co-prime.

(vi) 215 and 415
The factors of 215 are 1, 5, 43, 215
The factors of 415 are : 1, 5, 83, 415
Since, the common factor of 215 and 415 are 1 and 5.
∴ So, they are not co-prime.

Question 14.
Express each of the following numbers as the sum of two odd primes:
(i) 24
(ii) 36
(iii) 84
(iv) 98
Solution:
(i) 24
⇒ 24 = 5 + 19

(ii) 36
⇒ 36 = 7 + 29

(iii) 84
⇒ 84 = 17 + 67

(iv) 98
⇒ 98 = 19 + 79

Question 15.
Express each of the following numbers as the sum of twin-primes:
(i) 24
(ii) 36
(iii) 84
(iv) 120
Solution:
(i) 24
⇒ 24 = 11 + 13

(ii) 36
⇒ 36 = 17 + 19

(iii) 84
⇒ 84 = 41 + 43

(iv) 120
⇒ 120 = = 59 + 61

Question 16.
Express each of the following numbers as the sum of three odd primes:
(i) 21
(ii) 35
(iii) 49
(iv) 63
Solution:
(i) 21
⇒ 21 = 3 + 7 + 11

(ii) 35
⇒ 35 = 5 + 11 + 19

(iii) 49
⇒ 49 = 7 + 11 + 31

(iv) 63
⇒ 63 = 7 + 13 + 43