NCERT solutions for class 10 maths chapter 1, Real Numbers, split into two exercises: Exercise 1.1 on prime factorisation, HCF and LCM, and Exercise 1.2 on proving irrationality. This page routes you to the right exercise and tells you the method each one wants — the solutions themselves live one level down, on each exercise page.
The chapter runs on two big ideas. First, the Fundamental Theorem of Arithmetic: every composite number factorises uniquely as a product of primes, apart from order. Second, proof by contradiction, the technique that proves numbers like \( \sqrt{5} \) are irrational. Exercise 1.1 drills the first idea; Exercise 1.2 drills the second.
| Exercise | What you get |
|---|---|
| Exercise 1.1 | Solved |
| Exercise 1.2 | Solved |
Class 10 Maths Chapter 1 Exercises: What Each One Tests
Exercise 1.1 (7 questions) is the prime factorisation workout.
It opens with five composites to break into primes, then pairs and triples of numbers for HCF and LCM, a shortcut that uses a given HCF to reach the LCM, and two reasoning questions where you inspect a factorisation rather than compute: whether a power like \( 6^n \) can end with the digit zero, and why two built-up expressions are composite.
It closes with a circular-track word problem that is really an LCM question in disguise.
Exercise 1.2 (3 questions) is the proof section. It asks you to prove \( \sqrt{5} \) is irrational by contradiction, then extends the same argument to a sum form \( 3 + 2\sqrt{5} \), and finally to a reciprocal \( \frac{1}{\sqrt{2}} \), a rational multiple \( 7\sqrt{5} \), and another sum \( 6 + \sqrt{2} \). The same skeleton — assume rational, reduce to coprime form, square, apply Theorem 1.2, reach a contradiction — carries all of them.
Methods and Key Results for Real Numbers
This chapter runs on three techniques. Master these and both exercises open up.
1. Prime factorisation method (Fundamental Theorem of Arithmetic)
Every composite number factorises uniquely as a product of primes, apart from order. The factor tree on page 2 is the tool: it breaks a large number like 32760 into prime branches until only primes remain, giving \( 32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13 \).
From a factorisation you read off two results:
- HCF is the product of the smallest power of each common prime factor.
- LCM is the product of the greatest power of each prime factor involved.
2. The two-number identity
For any two positive integers \( a \) and \( b \):
\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \]
This is the shortcut behind Exercise 1.1 Q4: when the HCF is already given, the LCM follows directly as the product divided by the HCF.
3. Proof by contradiction for irrationality
To prove a number is irrational, assume the opposite — that it is rational, write it as \( \frac{p}{q} \) — then reduce the fraction to coprime form. Square both sides and apply Theorem 1.2: if a prime \( p \) divides \( a^2 \), then \( p \) divides \( a \). The steps force the prime to divide both numerator and denominator, contradicting coprimality. That contradiction is the proof.
Three-number caveat
The Note to the Reader on page 9 is a trap: for three numbers, the identity fails. Instead:
\[ \text{LCM}(p, q, r) = \frac{p \cdot q \cdot r \cdot \text{HCF}(p, q, r)}{\text{HCF}(p, q) \cdot \text{HCF}(q, r) \cdot \text{HCF}(p, r)} \]
\[ \text{HCF}(p, q, r) = \frac{p \cdot q \cdot r \cdot \text{LCM}(p, q, r)}{\text{LCM}(p, q) \cdot \text{LCM}(q, r) \cdot \text{LCM}(p, r)} \]
Common Mistakes in Real Numbers
- Applying \( \text{HCF} \times \text{LCM} = \text{product} \) to three numbers. It fails. Use the pairwise formulas from the Note to the Reader on page 9 instead.
- Skipping the coprime step in irrationality proofs. The contradiction depends on \( a \) and \( b \) having no common factor. If you never reduce the fraction, you cannot reach the contradiction.
- Confusing “ends with digit zero” with divisibility by 5. A number ends in zero only if its prime factorisation contains both 2 and 5. A power of 6 has only 2 and 3, so it can never end in zero.
- Treating the order of prime factors as two different factorisations. Uniqueness is apart from order — \( 2 \times 3 \times 5 \times 7 \) and \( 3 \times 5 \times 7 \times 2 \) are the same factorisation.
How to Use These Solutions for Board Preparation
Do Exercise 1.1 first. Prime factorisation and HCF/LCM are the foundation; without them, the proofs in Exercise 1.2 feel abstract. Then move to Exercise 1.2, where the reward is a reusable proof template.
Board questions favour two patterns: the “ends with digit zero” style (a divisibility-by-5 question) and the proof that \( \sqrt{p} \) is irrational for a prime \( p \). Practise both until the factorisation and the contradiction flow without looking.
For step-by-step working, open the Class 10 Maths notes hub or the Class 10 hub and follow the links to the chapter exercise solutions. To verify the original text and exercise numbers, check the official NCERT Class 10 Mathematics textbook.
| Exercise | What you get |
|---|---|
| Exercise 1.1 | Solved |
| Exercise 1.2 | Solved |
FAQs on Class 10 Maths Chapter 1 Real Numbers
Which exercise in Class 10 Maths Chapter 1 covers prime factorisation and HCF LCM?
Exercise 1.1. It has seven questions on prime factorisation, HCF, LCM, and the reasons behind divisibility results.
How do you prove that \( \sqrt{5} \) is irrational in Exercise 1.2?
By contradiction. Assume \( \sqrt{5} = \frac{a}{b} \) with \( a, b \) coprime, square both sides, apply Theorem 1.2 to force 5 to divide both, and reach a contradiction.
Does HCF × LCM equal the product of the numbers for three numbers?
No. The identity holds only for two numbers. For three numbers, use the pairwise HCF or LCM formulas given in the Note to the Reader.
What does it mean when a question asks if \( 6^n \) can end with the digit zero?
It is really a divisibility-by-5 question. A number ends in zero only if its prime factorisation contains both 2 and 5, so you inspect the factorisation of \( 6^n \) for the prime 5.
Is Exercise 1.1 or Exercise 1.2 more important for the board exam?
Both carry weight. Exercise 1.1 questions on HCF and LCM appear regularly, and proofs of irrationality from Exercise 1.2 are a standard board pattern.
Reference: NCERT textbooks (CBSE).