**NCERT Solutions Class 10 Maths Chapter 1 Real Numbers** are available here on this page. These solutions are provided to help students in their 2023-24 board exam preparations. These solved NCERT Class 10 Maths Real numbers questions will assist students in identifying the right approach to solve the problems quickly. These NCERT solutions of Class 10 maths chapter 1 are easy to understand for students. Detailed and step-wise explanations are given for every question of all exercises of the NCERT book (Class 10 Maths Chapter 1).

### Important Points about Real Numbers

**The Fundamental Theorem of Arithmetic :**

- Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
- If p is a prime and p divides a
^{2}, then p divides a, where a is a positive integer. - To prove that √2 and √3 are irrational.

### EXERCISE 1.1

1. Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15, and 21 (ii) 17, 23, and 29 (iii) 8, 9 and 25

4. Given that HCF (306, 657) = 9, find LCM (306, 657).

5. Check whether 6^{n} can end with the digit 0 for any natural number n.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

### EXERCISE 1.2

1. Prove that √5 is irrational.

2. Prove that 3 + 2√5 is irrational.

3. Prove that the following are irrationals :

(i) 1/√2 (ii) 7√5 (iii) 6 + √2

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