How do I find the HCF and LCM of two numbers?

In one line

The rule

The Highest Common Factor (HCF) of two numbers is the largest number that divides both without a remainder. The Lowest Common Multiple (LCM) is the smallest number that is a multiple of both numbers.

Formulas:

• HCF(a, b) = largest divisor of a and b
• LCM(a, b) = (a × b) / HCF(a, b)

Let's understand it simply

Imagine you and your friend are rolling rotis. You both want to cut your dough into equal pieces. HCF tells you the biggest size you can cut so both of you get equal pieces with nothing left. LCM tells you the smallest number of pieces you both can make so that both your doughs finish together, with no leftover.

How it works

1. List the factors of both numbers (factors are numbers that divide exactly).
2. Find the biggest common factor — this is the HCF.
3. List the multiples of both numbers (multiples are like 2×, 3×, 4× the number).
4. Find the smallest common multiple — this is the LCM.
5. Or, use prime factorisation: break both numbers into their prime factors, pick common ones for HCF, and all for LCM.

The story behind it

People have been finding HCF and LCM for thousands of years, even in ancient India and Greece. The famous Greek mathematician Euclid described a method to find HCF (now called the Euclidean algorithm) over 2,000 years ago. Merchants and cooks used these ideas to share things equally or to plan events that repeat after certain days.

Where you see it in real life

• Dividing sweets: You have 12 and 18 sweets. HCF tells you the biggest packet size you can make with no sweets left.
• Ringing bells: One bell rings every 4 minutes, another every 6 minutes. LCM tells you when both will ring together next.
• Buying packs: You want to buy pens in packs of 8 and 12, and want the same number of pens. LCM tells you the smallest number to buy.
• Cricket overs: Two bowlers bowl in 3 and 4 over spells. LCM tells after how many overs both will finish together.

Why it matters

• Sharing or grouping things equally: HCF helps you split things fairly.
• Planning events: LCM helps you find when things happen together.
• Maths exams: Many word problems use HCF and LCM.
• Later maths: Fractions, ratios, and algebra use these ideas.

Worked example

Let's find HCF and LCM of 12 and 18.

Given: 12 and 18
Step 1: List factors
  Factors of 12: 1, 2, 3, 4, 6, 12
  Factors of 18: 1, 2, 3, 6, 9, 18
Step 2: Find HCF
  Common factors: 1, 2, 3, 6
  HCF = 6
Step 3: Use formula for LCM
  LCM = (12 × 18) / 6
  LCM = 216 / 6
  LCM = 36

Common mistakes

• Mixing up HCF and LCM: Remember, HCF is about dividing, LCM is about multiplying.
• Missing factors or multiples: Always list all factors/multiples to avoid missing the correct answer.
• Forgetting to use the formula: For LCM, use (a × b) / HCF(a, b) to save time.

Remember it

HCF = Highest Divider, LCM = Lowest Multiplier. HCF cuts, LCM joins.