CBSE Class 10 Maths: Real Numbers — Euclid's Algorithm Notes 2026

Euclid's Division Algorithm

• For any two positive integers a and b: a = bq + r where 0 ≤ r < b (Division Lemma)
• To find HCF: apply division lemma repeatedly until remainder = 0; last non-zero divisor is HCF
• Example: HCF(135, 225) using Euclid's algorithm step by step

Fundamental Theorem of Arithmetic

• Every composite number can be expressed as a product of primes in a unique way (prime factorisation)
• HCF = product of common prime factors (lowest powers); LCM = product of all prime factors (highest powers)
• HCF × LCM = Product of two numbers (for exactly two numbers)

Irrational Numbers

• Numbers not expressible as p/q (q ≠ 0, p and q integers); non-terminating non-repeating decimals
• Proof that √2 is irrational: assume rational → contradiction (both p and q divisible by 2)
• √2, √3, √5, π, e are all irrational

Rational Numbers and Decimal Expansion

• p/q is terminating if denominator (in lowest terms) has only factors of 2 and/or 5
• Non-terminating recurring if denominator has prime factors other than 2 and 5
• Convert recurring decimal to fraction: multiply by 10ⁿ and subtract

HCF and LCM Applications

• Find the largest number that divides a and b with same remainder: HCF of (a−r) and (b−r)
• Find smallest number divisible by p and q: LCM(p, q)
• CBSE: three-number HCF/LCM problems using prime factorisation

Proving Irrationality

• Prove √p is irrational for prime p (standard CBSE method)
• Prove that sums like (2 + √3), 3√2, √5 − 2 are irrational
• Contrapositive proof: assume rational form p/q in lowest terms → contradiction

CBSE Exam Tips — Real Numbers

• Show all steps in Euclid's algorithm; division lemma must be stated
• Proofs of irrationality must use contradiction method clearly
• HCF × LCM property valid only for two numbers — common CBSE trap