✦ NCF 2023 ALIGNED • VISUAL LEARNING SERIES ✦
Chapter 1: Real Numbers
Prime Factorisation, HCF-LCM & Proofs of Irrationality
Curated by Akash Srivastva
Real Numbers include all Rational and Irrational numbers. This chapter uses the Fundamental Theorem of Arithmetic to prove the irrationality of √2, √3, √5 and to find HCF & LCM using prime factorisation.
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Fundamental Theorem of Arithmetic
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Theorem 1.1
Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order of the primes.
Example: 32760 = 2³ × 3² × 5 × 7 × 13
4ⁿ can never end with digit 0 because its only prime factor is 2.
4ⁿ can never end with digit 0 because its only prime factor is 2.
Proofs of Irrationality
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Theorems 1.2 & 1.3
Theorem 1.2: If a prime p divides a², then p divides a.
Proof that √2 is irrational (by contradiction):
Assume √2 = a/b (co-prime). Then 2b² = a² → 2 divides a → a=2c → b²=2c² → 2 divides b. Contradiction. Hence √2 is irrational.
Assume √2 = a/b (co-prime). Then 2b² = a² → 2 divides a → a=2c → b²=2c² → 2 divides b. Contradiction. Hence √2 is irrational.
Same method proves √3, √5 and √p (p prime) are irrational. Also 5−√3, 3√2, 6+√2 are irrational.
HCF & LCM Applications
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Prime Factorisation Method
HCF = product of lowest powers of common primes
LCM = product of highest powers of all primes involved.
6 = 2¹ × 3¹ | 20 = 2² × 5¹
HCF = 2 LCM = 60
HCF(a,b) × LCM(a,b) = a × b
HCF = 2 LCM = 60
HCF(a,b) × LCM(a,b) = a × b
This relation holds only for two numbers.
Rational & Irrational Properties
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Key Properties
• Rational ± Irrational = Irrational
• Non-zero Rational × Irrational = Irrational
• Rational ÷ Irrational = Irrational
Examples:
5 − √3 is irrational
3√2 is irrational
(proved by contradiction)
5 − √3 is irrational
3√2 is irrational
(proved by contradiction)