HCF and LCM: methods, fractions and solved examples

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Definitions

HCF (highest common factor) is the largest number that divides all the given numbers. LCM (lowest common multiple) is the smallest number that all of them divide.

Prime factorisation method

Write each number as a product of primes. HCF = product of common primes with the lowest powers. LCM = product of all primes with the highest powers.

12 = 2² × 3, 18 = 2 × 3² → HCF = 2 × 3 = 6, LCM = 2² × 3² = 36.

Key rule (two numbers only)

Fractions

Remainder questions

• The largest number that divides a, b, c leaving the same remainder = HCF of (a − b), (b − c), (c − a).
• The smallest number that leaves remainder r when divided by a, b, c = LCM(a, b, c) + r.

Solved examples

1. The HCF of two numbers is 12 and their LCM is 360. One number is 72. Find the other.
   12 × 360/72 = 60.
2. Find the smallest number that leaves remainder 3 when divided by 6, 8 and 12.
   LCM = 24 → 24 + 3 = 27.
3. Three bells ring every 6, 10 and 15 minutes. After how long do they ring together again?
   LCM = 30 minutes.