Chapter 1 · Class X Mathematics · NCERT Exercises

Real Numbers — Exercises

Euclid to Irrationals — Every Real Numbers Exercise Fully Solved

📂 2 Exercises 📝 10 Questions 🎓 Foundation

Exercise Index

Euclid's Division Lemma; HCF using Euclid's algorithm; Fundamental Theorem of Arithmetic; LCM and HCF by prime factorisation

5 Q Easy-Moderate

②

Exercise 1.2

Irrational numbers; proof by contradiction; rational numbers; terminating vs recurring decimals

5 Q Moderate

2 exercise files · 10 total questions

Chapter at a Glance

CBSE BoardsNTSEOlympiad

7 Concepts

8 Formulas

Foundation Difficulty

6–8% Weightage

Before You Begin

Prerequisites

✓Number system basics (Class IX)
✓Prime and composite numbers
✓Basic division algorithm

Have Ready

🔧NCERT Class X textbook
🔧Factor tree diagram paper
🔧Notebook for proofs

Exercise Topic Map

Exercise 1.1 Apply a = bq + r; compute HCF(a,b) step-by-step using repeated division

Exercise 1.2 Express n as product of prime powers; HCF = product of common primes with least powers; LCM = product with highest powers

Key Formulae — Recall Before Solving

\(a = bq + r, \quad 0 \le r < b \quad \text{(Euclid's Division Lemma)}\)

\(\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b\)

\(\sqrt{p} \text{ is irrational for every prime } p\)

\(\frac{p}{q} \text{ terminates} \iff q = 2^m \cdot 5^n,\ m,n \in \mathbb{W}\)

NCERT Solving Method

Step 1 — Euclid's algorithm: apply a = bq+r repeatedly until r = 0; last non-zero remainder = HCF. Step 2 — FTA factorisation: write each number as product of primes using factor trees. Step 3 — Irrational proofs: assume √p = a/b in lowest terms; square both sides; derive contradiction via FTA. Step 4 — Decimal type: check denominator after full simplification; count only 2s and 5s.