Definitions
- HCF (or GCD): Highest Common Factor — the largest number that divides all given numbers exactly
- LCM: Least Common Multiple — the smallest number divisible by all given numbers
Key Relationship
HCF × LCM = Product of two numbers (valid only for exactly two numbers)
- HCF ≤ LCM always
- HCF is always a factor of LCM
Methods to Find HCF
- Prime Factorisation Method: Break each number into prime factors; HCF = product of common prime factors at their lowest power
- Division Method: Divide the larger number by the smaller; keep dividing the previous divisor by the remainder until remainder is 0 — the last divisor is the HCF
Methods to Find LCM
- Prime Factorisation Method: LCM = product of all prime factors at their highest power across all numbers
- Division Method: Divide all numbers together by common prime factors repeatedly; LCM = product of all divisors and remaining numbers
Applications
- LCM is used for problems involving events repeating at intervals (bells ringing together, traffic lights)
- HCF is used for problems involving dividing things into the largest equal groups (max tiles, max rope pieces)
💡 Example: HCF and LCM of 36 and 60 → 36=2²×3², 60=2²×3×5 → HCF=2²×3=12, LCM=2²×3²×5=180. Check: 12×180=2160=36×60 ✓