Numerical aptitude · Numbers

HCF, LCM and divisibility, made easy

💡 The idea

HCF = the biggest number that divides both exactly. LCM = the smallest number both divide into. “Biggest piece” → HCF. “When together again” → LCM.

✂️ Cut 12 m and 18 m into equal pieces
longest piece = HCF = 6→🔔 Bells every 4 and 6 min
together again = LCM = 12

Which one do I need?

🧭 The method: 3 steps

Read the question: biggest / longest / most that fits → HCF. Next time together / smallest that both fit → LCM.
HCF: write each number as factors and keep what they share. LCM: count up in the bigger number until the smaller one fits.
Check: HCF × LCM = the two numbers multiplied.

🎬 Every kind of question, in pictures

Pick a kind. Guess the answer first, then watch the picture work it out. Tap any step to see it in the picture.

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✏️ Worked examples

Example 1. HCF of 24 and 36?

24 = 12 × 2, 36 = 12 × 3
They share 12, and 2 and 3 share nothing more
HCF = 12

Example 2. LCM of 6 and 8?

Count in 8s: 8, 16, 24
Does 6 go into 8? No. 16? No. 24? Yes
LCM = 24

⚡ Shortcut: 6 × 8 ÷ HCF 2 = 24.

Example 3. Is 4,725 divisible by 9?

Add the digits: 4 + 7 + 2 + 5 = 18
18 divides by 9, so yes

⚠️ Where people go wrong

Multiplying the two numbers gives A common multiple, not always the LOWEST one.
Mixing up HCF and LCM: the HCF is never bigger than the smaller number; the LCM is never smaller than the bigger one.
For “same remainder” questions: LCM, then ADD the remainder.

📌 Remember

HCF ≤ smaller number ≤ bigger number ≤ LCM
HCF × LCM = a × b
÷3, ÷9: add the digits
Last digits of powers repeat every 4