HCF and LCM: Methods and Word Problems

HCF and LCM

HCF (Highest Common Factor) is the largest number that divides all given numbers exactly. LCM (Lowest Common Multiple) is the smallest number that is exactly divisible by all given numbers.

Methods

1. Prime factorisation: 12 = 2² × 3, 18 = 2 × 3². HCF = lowest powers = 2 × 3 = 6. LCM = highest powers = 2² × 3² = 36.
2. Division method (HCF): divide the larger by the smaller, then the divisor by the remainder, until remainder is 0.

Key rule

HCF × LCM = Product of two numbers (only for two numbers). Example: 6 × 36 = 12 × 18 = 216.

Word-problem signals (RRB NTPC)

• "Largest size / greatest number that divides" → HCF (e.g., cutting ropes into equal largest pieces).
• "Smallest number / when will they meet again / bells ring together" → LCM.
• Remainder type: smallest number which leaves remainder r when divided by a, b, c = LCM(a, b, c) + r.

Fractions

HCF of fractions = HCF of numerators ÷ LCM of denominators. LCM of fractions = LCM of numerators ÷ HCF of denominators.

Analogy

HCF is the biggest common spoon; LCM is the first common meeting point. If you must measure 12 litres and 18 litres of milk using the same jug with no leftovers, the biggest jug that works is 6 litres — that's HCF. If two buses leave the depot every 12 and 18 minutes, the first time they leave together again is after 36 minutes — that's LCM.