HCF and LCM Cheat Sheet

HCF and LCM Cheat Sheet

Quick Revision

The HCF (Highest Common Factor) of two or more numbers is the largest number that divides each of them exactly. The LCM (Lowest Common Multiple) is the smallest number that each of them divides exactly.

Ready to practise? Take the HCF and LCM quiz

1. Finding HCF and LCM by Prime Factorisation

Write each number as a product of primes.

• HCF = product of the common prime factors, each to its lowest power.
• LCM = product of all prime factors, each to its highest power.

Example

12 = 2² × 3 and 18 = 2 × 3².

HCF = 2¹ × 3¹ = 6
LCM = 2² × 3² = 36

2. The Product Rule (two numbers only)

For any two numbers:

HCF × LCM = Product of the two numbers

Example

For 12 and 18: HCF × LCM = 6 × 36 = 216 = 12 × 18. ✓

This rule holds for two numbers only, not for three or more.

3. HCF and LCM of Fractions

HCF of fractions = HCF of the numerators / LCM of the denominators
LCM of fractions = LCM of the numerators / HCF of the denominators

Example

For 3/4 and 9/10:

HCF = HCF(3, 9) / LCM(4, 10) = 3 / 20
LCM = LCM(3, 9) / HCF(4, 10) = 9 / 2

4. Common Word Problems

Bells example: bells ring every 6, 8 and 12 seconds; they ring together again after LCM(6, 8, 12) = 24 seconds.

5. Exam Quick Tips

• HCF ≤ each number ≤ LCM — a quick sanity check.
• The product rule works for two numbers only.
• "Greatest / largest that divides" → HCF; "least / together again" → LCM.
• For fractions, HCF uses LCM of denominators, and LCM uses HCF of denominators — do not swap them.

Keep revising with the rest of the Maths section:

• All Maths topics
• Number System Cheat Sheet
• Average Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.