LCM & HCF of 3 Numbers Using Prime Factorisation

 

❓ Question

Find the LCM and HCF of 6, 72 and 120 using prime factorisation method.

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🖼️ Solution Image

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✍️ Short Solution

This is a 3-number prime factorisation question.
Use:

• HCF → smallest powers of common factors
• LCM → greatest powers of all factors 💯

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🔹 Step 1 — Prime Factorisation

$$6=2^1\times 3^1$$
$$72=2^3\times 3^2$$
$$120=2^3\times 3^1\times 5^1$$

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🔹 Step 2 — Find HCF

👉 Common prime factors = 2 and 3

$$\text{HCF}=2^{\min (1,3,3)}\times 3^{\min (1,2,1)}$$
$$\text{HCF}=2^1\times 3^1=6$$

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🔹 Step 3 — Find LCM

👉 Take highest powers of all prime factors:

$$\text{LCM}=2^3\times 3^2\times 5^1$$
$$\text{LCM}=8\times 9\times 5=360$$

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🔹 Step 4 — Important Note

For three numbers:

$$𝑎\times 𝑏\times 𝑐\mathrm{\ne }\text{HCF}\times \text{LCM}$$

📌 This relation works only for two numbers

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✅ Final Answer

$$\boxed{{\displaystyle \text{HCF}=6,\text{ LCM}=360}}$$

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⭐ Key Insight

• HCF → common minimum powers
• LCM → maximum powers of all primes

🧠 Memory Line:

HCF = lowest common, LCM = highest overall