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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🧱 Factors — they divide the number
A factor divides the number leaving no remainder.
Factors of 12: 1, 2, 3, 4, 6, 12
🔁 Multiples — the number's table
A multiple is what you get when you multiply the number.
Multiples of 12: 12, 24, 36, 48, 60, …
🌳 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.
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.
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 ✂️
Use LCM when things REPEAT 🔁
12 AISSEE-style questions on factors, HCF, LCM and word problems. 🍀
