Prime factorisation provides an efficient and systematic method for computing both the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two or more numbers. This topic is central to the CBSE Class 10 Real Numbers chapter and has practical applications in scheduling, resource distribution, and problem solving.
To find HCF using prime factorisation: (1) Write each number as a product of prime factors. (2) Identify all prime factors that appear in every factorisation. (3) For each common factor, choose the smallest power that appears. (4) Multiply these together. Example: HCF(12, 18) — 12 = 2² × 3, 18 = 2 × 3². Common factors: 2¹ and 3¹. HCF = 2 × 3 = 6.
To find LCM using prime factorisation: (1) Write each number as a product of prime factors. (2) List every prime factor that appears in any factorisation. (3) For each factor, choose the greatest power that appears. (4) Multiply these together. Example: LCM(12, 18) — 12 = 2² × 3, 18 = 2 × 3². All factors with greatest powers: 2² and 3². LCM = 4 × 9 = 36.
For two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. This formula is extremely useful when one of the four quantities is unknown. Note: this relationship does not extend directly to three or more numbers.
HCF is used when dividing resources equally — for example, cutting fabric of different lengths into the largest possible equal pieces. LCM is used for synchronisation — for example, determining when two events with different cycles will coincide again (e.g., two traffic lights, bus schedules, or gear rotations).
Use HCF when you need the largest common measure or equal distribution. Use LCM when you need the smallest common multiple or synchronisation point.
No, HCF(a, b, c) × LCM(a, b, c) ≠ a × b × c in general. The simple product relationship applies only to two numbers.
For two numbers, Euclid's Division Algorithm is often faster than prime factorisation. For three or more numbers, prime factorisation is usually more systematic.
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