What types of questions does Exercise 5.4 have in Chapter 5?

Exercise 5.4 contains puzzles or real-life problems involving factors, multiples and prime numbersA.

How does this exercise assess students' knowledge of the previous content?

It requires students to make use of ideas from previous exercise content such as H.C.F, L.C.M and prime factorisation.

How do these questions improve the knowledge for the students in preparation for Olympiads?

It develops a strong set of problem-solving skills, which helps in being able to answer the maths Olympiad questions quickly.

What are the big learning advantages of working with the NCERT solutions?

These NCERT solutions help improve the critical thinking process, help with being exam ready, and make difficult problems easier to tackle.

NCERT Class 6 Maths Ch 5 Exercise 5.4 Solutions

Find the prime factorisations of the following numbers:

64, 104, 105, 243, 320, 141, 1728

, 729, 1024, 1331, 1000. Sol. (1) The prime factorisation of 64 is

2 \times 2 \times 2 \times 2 \times 2 \times 2

. (2) The prime factorisation of 104 is

2 \times 2 \times 2 \times 13

. (3) The prime factorisation of 105 is

3 \times 5 \times 7

. (4) The prime factorisation of 243 is

3 \times 3 \times 3 \times 3 \times 3

. (5) The prime factorisation of 320 is

2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5

. (6) The prime factorisation of 141 is

3 \times 47

. (7) The prime factorisation of 1728 is

2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3

. (8) The prime factorisation of 729 is

3 \times 3 \times 3 \times 3 \times 3 \times 3

. (9) The prime factorisation of 1024 is

2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2

. (10) The prime factorisation of 1331 is

11 \times 11 \times 11

. (11) The prime factorisation of 1000 is

2 \times 2 \times 2 \times 5 \times 5 \times 5

.

The prime factorisation of a number has one 2 , two 3 s , and one 11 . What is the number? Sol. To find the number, we multiply these prime factors together:

2 \times 3 \times 3 \times 11 = 198

Thus, the number is 198.

Find three prime numbers, all less than 30 , whose product is 1955 . Sol. The prime factorisation of 1955: 1955

= 5 \times 17 \times 23

. All the factors are prime numbers and are less than 30 . Hence, the three prime numbers whose product is 1955 are 5, 17, and 23.

Find the prime factorisation of these numbers without multiplying first. (a)

56 \times 25

(b)

108 \times 75

(c)

1000 \times 81

Sol. (a) Prime factors of

56 = 2 \times 2 \times 2 \times 7

Prime factors of

25 = 5 \times 5

Combined prime factorisation of

56 \times 25 = 2 \times 2 \times 2 \times 7 \times 5 \times 5

(b) Prime factors of

108 = 2 \times 2 \times 3 \times 3 \times 3

Prime factors of

75 = 3 \times 5 \times 5

Combined prime factorisation of

108 \times 75 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5

(c) Prime factors of

1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5

Prime factors of

81 = 3 \times 3 \times 3 \times 3

Combined prime factorisation of

1000 \times 81 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 3 \times 3 \times 3 \times 3

What is the smallest number whose prime factorisation has: (a) three different prime numbers? (b) four different prime numbers? Sol. (a) The smallest prime numbers are 2,3 , and 5 . To find the smallest number with these primes as factors, multiply them together:

2 \times 3 \times 5 = 30

. So, the smallest number whose prime factorisation has three different prime numbers is 30 . (b) The smallest four prime numbers are

2, 3, 5

, and 7 . To find the smallest number with these primes as factors, multiply them together:

2 \times 3 \times 5 \times 7 = 210

. Thus, the smallest number whose prime factorisation has four different prime numbers is 210.

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. (a) 30 and 45 (b) 57 and 85 (c) 121 and 1331 (d) 343 and 216 Sol. (a) Factors of 30 and 45:

30 = 2 \times 3 \times 5 45 = 3 \times 3 \times 5

Common factors:

3 \times 5 = 15

. Hence, 30 and 45 are not a pair of co-prime numbers. (b) Factors of 57 and 85:

57 = 3 \times 19

85 = 5 \times 17

No common factors other than 1 . Hence 57 and 85 are a pair of co-prime numbers. (c) Factors of 121 and 1331:

121 = 11 \times 11

1331 = 11 \times 11 \times 11

Common factors:

11 \times 11 = 121

. Hence 121 and 1331 are not a pair of co-prime numbers. (d) Factors of 343 and 216:

343 = 7 \times 7 \times 7

216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3

No common factors other than 1 . Hence 343 and 216 are a pair of co-prime numbers.

Is the first number divisible by the second? Use prime factorisation. (a) 225 and 27 (b) 96 and 24 (c) 343 and 17 (d) 999 and 99 Sol. (a) Prime Factors of 225 and 27:

225 = 3 \times 3 \times 5 \times 5

and

27 = 3 \times 3 \times 3

Since 225 contains

3 \times 3

and does not have enough factors of 3 to match

3 \times 3 \times 3, 225

does not have sufficient factors to be divisible by 27. Therefore, 225 is not divisible by 27. (b) Prime Factors of 96 and 24:

96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3

and

24 = 2 \times 2 \times 2 \times 3

Since 96 includes the required factors to match those in 24 , it is divisible by 24 . (c) Prime Factors of 343 and 17:

343 = 7 \times 7 \times 7

and

17 = 1 \times 17

Since the prime factorisation of 343 contains the prime factor 7 but not 17. Thus, 343 is not divisible by 17 . (d) Prime Factors of 999 and 99:

999 = 3 \times 3 \times 3 \times 37

and

99 = 3 \times 3 \times 11

Since 999 does not include the factor 11, which is required for divisibility by 99 . Hence, 999 is not divisible by 99.

The first number has prime factorisation

2 \times 3 \times 7

and the second number has prime factorisation

3 \times 7 \times 11

. Are they co-prime? Does one of them divide the other? Sol. The numbers share the common factors 3 and 7. So they are not co-prime since neither number contains all the factors of the other, neither can divide the other.

Guna says, "Any two prime numbers are co-prime". Is he right? Sol. Yes, Guna is right. Any two prime numbers are co-prime as they do not have common factor other than 1 which means they are always co-prime. For example, 2 and 3,5 and 7,11 and 13 .

Frequently Asked Questions

What types of questions does Exercise 5.4 have in Chapter 5?

How does this exercise assess students' knowledge of the previous content?

How do these questions improve the knowledge for the students in preparation for Olympiads?

What are the big learning advantages of working with the NCERT solutions?

Join ALLEN!

NCERT Solutions Class 6 Maths Chapter 5 Prime Time Exercise 5.4

1.0Download NCERT Solutions Class 6 Maths Chapter 5 Prime Time Exercise 5.4: Free PDF

2.0Key Concepts Covered in Exercise 5.4 Class 6 Maths

3.0NCERT Solutions Class 6 Chapter 5 Prime Time: All Exercises

4.0NCERT Class 6 Maths Chapter 5 Prime Time Exercise 5.4: Detailed Solutions

5.0Key Features and benefits for Class 6 Maths Chapter 5 Exercise 5.4

Frequently Asked Questions

What types of questions does Exercise 5.4 have in Chapter 5?

How does this exercise assess students' knowledge of the previous content?

How do these questions improve the knowledge for the students in preparation for Olympiads?

What are the big learning advantages of working with the NCERT solutions?

Join ALLEN!

NCERT Solutions Class 6 Maths Chapter 5 Prime Time Exercise 5.4

1.0Download NCERT Solutions Class 6 Maths Chapter 5 Prime Time Exercise 5.4: Free PDF

2.0Key Concepts Covered in Exercise 5.4 Class 6 Maths

3.0NCERT Solutions Class 6 Chapter 5 Prime Time: All Exercises

4.0NCERT Class 6 Maths Chapter 5 Prime Time Exercise 5.4: Detailed Solutions

5.0Key Features and benefits for Class 6 Maths Chapter 5 Exercise 5.4