Mathematics · Class 10
Real Numbers
Euclid's division lemma, HCF-LCM via primes, and irrationality proofs.
3 sections15–25 min read5 MCQs · 2 examples
Chapter Overview
Class 10 Real Numbers revisits Euclid's division lemma, fundamental theorem of arithmetic (prime factorisation), and revisits rational/irrational numbers — a high-weight board chapter.
Euclid's Division Lemma
For positive integers a and b, ∃ unique q, r such that a = bq + r, 0 ≤ r < b.
Fundamental Theorem of Arithmetic
- Every composite number factors uniquely into primes (order ignored)
- HCF × LCM = product of two numbers
- Use prime factorisation to find HCF and LCM
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: HCF by Euclid
Find HCF of 135 and 225.
- 1225 = 135×1 + 90
- 2135 = 90×1 + 45
- 390 = 45×2 + 0
- 4HCF = 45
Answer
45
Example 2: LCM via primes
Find LCM of 12 and 18.
- 112 = 2²×3
- 218 = 2×3²
- 3LCM = 2²×3² = 36
Answer
36
Key Points to Remember
- ✓ HCF is product of smallest powers of common primes
- ✓ LCM is product of greatest powers of all primes
- ✓ √p is irrational if p is prime
Exam Tips
- • Show Euclid steps for full marks
- • Prove irrationality using contradiction method
Practice MCQs — Real Numbers
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 5Score: 0/0
HCF of 96 and 404 by Euclid is: