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SECTION A · MATHEMATICS · TOPIC 2

LCM and HCF ➗

Two words, one idea: numbers share factors and multiples. AISSEE asks these almost every year — usually hidden inside a word problem about bells, tiles or rows of soldiers.

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Part 1 · Factors & Multiples

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Part 2 · HCF & LCM

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Part 3 · Word Problems

1

Factors & Multiples 🧱

🧱 Factors — they divide the number

A factor divides the number leaving no remainder.

Factors of 12: 1, 2, 3, 4, 6, 12

• Every number's smallest factor is 1, biggest is itself.
• A number has a limited number of factors.

🔁 Multiples — the number's table

A multiple is what you get when you multiply the number.

Multiples of 12: 12, 24, 36, 48, 60, …

• The smallest multiple of a number is itself.
• Multiples are endless.

🌳 Prime factorisation — break it into primes

Keep dividing by the smallest prime that fits (2, then 3, then 5, 7, 11 …) until you reach 1.

Factorise 60

2 | 60

2 | 30

3 | 15

5 | 5

| 1

60 = 2 × 2 × 3 × 5

Factorise 72

2 | 72

2 | 36

2 | 18

3 | 9

3 | 3

| 1

72 = 2 × 2 × 2 × 3 × 3

Divisibility shortcuts: by 2 if it ends in an even digit · by 3 if the digit sum is in the 3 table · by 5 if it ends in 0 or 5 · by 9 if the digit sum is in the 9 table · by 10 if it ends in 0.

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Game 1 — Prime or Composite?

2

HCF & LCM 🤝

HCF ⬇️

Highest Common Factor

The biggest number that divides all of them. It is always smaller than or equal to the numbers.

Factors of 12: 1,2,3,4,6,12
Factors of 18: 1,2,3,6,9,18
HCF = 6

LCM ⬆️

Lowest Common Multiple

The smallest number that all of them divide into. It is always bigger than or equal to the numbers.

Multiples of 12: 12,24,36,48…
Multiples of 18: 18,36,54…
LCM = 36

⚡ The fast method — prime factors

For HCF — take what is COMMON

Circle the primes that appear in every number, using the lowest power. Multiply them.

For LCM — take EVERYTHING

Take every prime that appears anywhere, using the highest power. Multiply them.

Example — 60 and 72

60 = 2 × 2 × 3 × 5 | 72 = 2 × 2 × 2 × 3 × 3

Common: two 2s and one 3 → HCF = 2 × 2 × 3 = 12

All, highest power: 2×2×2 (from 72), 3×3 (from 72), 5 (from 60) → LCM = 8 × 9 × 5 = 360

Golden formula (for two numbers only): HCF × LCM = first number × second number.
Check: 12 × 360 = 4320, and 60 × 72 = 4320 ✅ — a brilliant way to verify your answer.

📘 Solved for you, step by step

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Game 2 — Find the HCF and LCM

3

Word Problems 📖

The hard part is not the sum — it is deciding which one to use. Here is the rule that never fails.

Use HCF when things get SPLIT ✂️

• "greatest / largest / maximum size"
• cutting rope or cloth into equal pieces
• making the largest square tiles
• dividing sweets equally with none left
• the biggest answer to a dividing question

Use LCM when things REPEAT 🔁

• "least / smallest / minimum number"
• bells ringing together again
• runners meeting at the start line
• buses leaving at the same time
• the smallest answer to a repeating question

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Game 3 — HCF or LCM?

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Topic Mock Test

12 AISSEE-style questions on factors, HCF, LCM and word problems. 🍀

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Maths 3 · Unitary Method — coming soon 🔒