Prime Factorisation Method for LCM & HCF

 

❓ Question

Find the LCM and HCF of 6 and 20 by prime factorisation method.

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🖼️ Solution Image

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✍️ Short Solution

This is a prime factorisation-based question.
Use:

• HCF → smallest powers
• LCM → greatest powers 💯

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🔹 Step 1 — Prime Factorisation

$$6 = 2^1 \times 3^1$$
$$20 = 2^2 \times 5^1$$

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🔹 Step 2 — Find HCF

👉 HCF = product of smallest powers of common prime factors

Common prime factor = 2

$$\text{HCF} = 2^1 = 2$$

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🔹 Step 3 — Find LCM

👉 LCM = product of greatest powers of all prime factors

$$\text{LCM} = 2^2 \times 3^1 \times 5^1$$
$$\text{LCM} = 4 \times 3 \times 5 = 60$$

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🔹 Step 4 — Verification

$$\text{HCF} \times \text{LCM} = 2 \times 60 = 120$$
$$6 \times 20 = 120$$

👉 Verified ✅

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✅ Final Answer

$$\boxed{\text{HCF} = 2,\ \text{LCM} = 60}$$

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⭐ Key Insight

• HCF → common factors (minimum power)
• LCM → all factors (maximum power)

🧠 Memory Line:

HCF = common lowest, LCM = total highest