Class 10 Mathematics

Chapter 1 Solutions — Real Numbers

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Solutions

Overview

Step-by-step NCERT solutions for Real Numbers (Chapter 1, CBSE Class 10 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Real Numbers textbook chapter.

Solved

What these solutions cover

All 10 questions in Real Numbers are solved in the PDF. Here's what's inside, exercise by exercise:

Fundamental Theorem of Arithmetic — HCF & LCM

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
Given that HCF (306, 657) = 9, find LCM (306, 657).
Check whether 6ⁿ can end with the digit 0 for any natural number n.
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
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?

Irrational Numbers — proofs

Prove that √5 is irrational.
Prove that 3 + 2√5 is irrational.
3.
Prove that the following are irrationals:
- (i) 1/√2
- (ii) 7√5
- (iii) 6 + √2

Board papers

Real Numbers in CBSE board papers

Tested in 83 questions across 23 Class 10 board papers (2020–2026). Open a set to see exactly how this chapter is asked:

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