NCERT Solutions for Class 6 Maths Chapter 5 - Prime Time Exercise 5.4

Exercise 5.4

Figure it Out 

1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 

Ans:

• 64 = $ 2^6 $

• 104 = $ 2^3 \times 13 $

• 105 = $ 3 \times 5 \times 7 $

• 243 = $ 3^5 $

• 320 = $ 2^6 \times 5 $

• 141 = $ 3 \times 47 $

• 1728 = $ 2^6 \times 3^3 $

• 729 = $ 3^6 $

• 1024 = $ 2^{10} $

• 1331 = $ 11^3 $

• 1000 = $ 2^3 \times 5^3 $

2. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number? 

Ans:

• The prime factorisation is $ 2 \times 3^2 \times 11 $.

• The number is: $ 2 \times 9 \times 11 = 198 $.

• The number is 198.

3. Find three prime numbers, all less than 30, whose product is 1955. 

Ans:

• Prime factorisation of 1955: 

$ 1955 \div 5 = 391 $ 

$ 391 \div 17 = 23 $.

• The three prime numbers are: 

$ 5 \times 17 \times 23 = 1955 $.

• The three prime numbers are 5, 17, and 23.

4. Find the prime factorisation of these numbers without multiplying first 

a. 56 × 25 

b. 108 × 75 

c. 1000 × 81 

Ans:

a. 56 × 25 

• Prime factorisation of 56 = $ 2^3 \times 7 $ 

• Prime factorisation of 25 = $ 5^2 $ 

• Prime factorisation of 56 × 25 = $ 2^3 \times 7 \times 5^2 $

b. 108 × 75 

• Prime factorisation of 108 = $ 2^2 \times 3^3 $ 

• Prime factorisation of 75 = $ 3 \times 5^2 $ 

• Prime factorisation of 108 × 75 = $ 2^2 \times 3^4 \times 5^2 $

c. 1000 × 81 

• Prime factorisation of 1000 = $ 2^3 \times 5^3 $ 

• Prime factorisation of 81 = $ 3^4 $ 

• Prime factorisation of 1000 × 81 = $ 2^3 \times 5^3 \times 3^4 $

5. What is the smallest number whose prime factorisation has: 

a. three different prime numbers? 

b. four different prime numbers?  

Ans:

a. Three different prime numbers: 

• The smallest three different primes are 2, 3, and 5. 

• The smallest number = $ 2 \times 3 \times 5 = 30 $.

b. Four different prime numbers: 

• The smallest four different primes are 2, 3, 5, and 7. 

• The smallest number = $ 2 \times 3 \times 5 \times 7 = 210 $.

Figure it Out 

1. Are the following pairs of numbers coprime? 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 

Ans: Two numbers are co-prime if their greatest common divisor (GCD) is 1. We will use prime factorisation to check this.

a. 30 and 45 

• Prime factorisation of 30: $ 2 \times 3 \times 5 $ 

• Prime factorisation of 45: $ 3^2 \times 5 $ 

• Common factors: 3 and 5 

• Not coprime because they share common factors (3 and 5).

b. 57 and 85

• Prime factorisation of 57: $ 3 \times 19 $ 

• Prime factorisation of 85: $ 5 \times 17 $ 

• No common factors. 

• Co-prime because they have no common factors.

c. 121 and 1331 

• Prime factorisation of 121: $ 11^2 $ 

• Prime factorisation of 1331: $ 11^3 $ 

• Common factor: 11 

• Not coprime because they share a common factor (11).

d. 343 and 216 

• Prime factorisation of 343: $ 7^3 $ 

• Prime factorisation of 216: $ 2^3 \times 3^3 $ 

• No common factors. 

• Co-prime because they have no common factors.

2. 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 

Ans:

a. 225 and 27 

• Prime factorisation of 225: $ 3^2 \times 5^2 $ 

• Prime factorisation of 27: $ 3^3 $ 

• 225 is not divisible by 27 because $ 3^3 $ is not a factor of 225 (it only has $ 3^2 $).

b. 96 and 24 

• Prime factorisation of 96: $ 2^5 \times 3 $ 

• Prime factorisation of 24: $ 2^3 \times 3 $ 

• 96 is divisible by 24 because all the prime factors of 24 are present in 96.

c. 343 and 17 

• Prime factorisation of 343: $ 7^3 $ 

• Prime factorisation of 17: 17 is a prime number, not a factor of 343. 

• 343 is not divisible by 17.

d. 999 and 99 

• Prime factorisation of 999: $ 3^3 \times 37 $ 

• Prime factorisation of 99: $ 3^2 \times 11 $ 

• 999 is not divisible by 99 because it lacks the factor 11.

3. The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? Does one of them divide the other? 

Ans:

• Prime factorisation of the first number: $ 2 \times 3 \times 7 $

• Prime factorisation of the second number: $ 3 \times 7 \times 11 $

• Common factors: 3 and 7

They are not coprime because they share common factors (3 and 7). 

Neither divides the other because the prime factorisations are different (the first number has 2, the second has 11).

4. Guna says, “Any two prime numbers are co-prime?”. Is he right?

Ans: Yes, Guna is right. 

Any two prime numbers are always coprime because prime numbers have no common factors other than 1. For example, 3 and 5 are both prime numbers, and they have no common factors except for 1.

Benefits of NCERT Solutions for Class 6 Maths Chapter 5 Exercise 5.4 Prime Time

• Clear Understanding of Prime and Composite Numbers: The solutions explain the concepts of prime and composite numbers in a simple and easy-to-understand manner.

• Step-by-Step Explanations: Each problem in Exercise 5.4 is solved step-by-step, helping students learn how to approach and solve problems systematically.

• Improves Number Theory Skills: The exercise strengthens students' number theory knowledge by teaching them about divisibility rules and factorisation.

• Aligned with CBSE Syllabus: The solutions are fully aligned with the CBSE Class 6 Maths syllabus, ensuring that students cover all essential topics.

• Boosts Exam Preparation: These solutions provide ample practice, making it easier for students to revise and prepare confidently for exams.

• Free PDF Access: Students can download the NCERT Solutions in FREE PDF format, allowing for easy access and convenient study.

Class 6 Maths Chapter 5: Exercises Breakdown

Important Study Material Links for Class 6 Maths Chapter 5 - Prime Time

Conclusion

The NCERT Solutions for Class 6 Maths Chapter 5 Exercise 5.4, Prime Time, offer a clear and structured way for students to master the concepts of prime and composite numbers. With step-by-step explanations, these solutions simplify the process of learning divisibility rules and factorisation. Aligned with the CBSE syllabus, they serve as an excellent tool for exam preparation and regular practice. Available for FREE PDF download, these solutions provide a convenient and effective way for students to strengthen their number theory skills and boost their confidence in solving mathematical problems.

Chapter-wise NCERT Solutions Class 6 Maths

The chapter-wise NCERT Solutions for Class 6 Maths are given below. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

Related Important Links for Class 6  Maths 

Along with this, students can also download additional study materials provided by Vedantu for Maths Class 6.