HCF & LCM of Three Numbers Using Prime Factorisation

 

❓ Question

Find the LCM and HCF of the following integers by applying the prime factorisation method:

(i) 12, 15 and 21
(ii) 17, 23 and 29
(iii) 8, 9 and 25

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🖼️ Solution Image

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✍️ Short Solution

This is a prime factorisation question involving three numbers.

Use:

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

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🔹 (i) 12, 15 and 21

Prime Factorisation

$$12 = 2^2 \times 3$$
$$15 = 3 \times 5$$
$$21 = 3 \times 7$$

HCF

Common factor = 3

$$\text{HCF} = 3$$

LCM

$$\text{LCM} = 2^2 \times 3 \times 5 \times 7$$
$$\text{LCM} = 420$$

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🔹 (ii) 17, 23 and 29

Prime Factorisation

All three numbers are prime numbers.

$$17 = 17$$
$$23 = 23$$
$$29 = 29$$

HCF

No common factor except 1.

$$\text{HCF} = 1$$

LCM

$$\text{LCM} = 17 \times 23 \times 29$$
$$\text{LCM} = 11339$$

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🔹 (iii) 8, 9 and 25

Prime Factorisation

$$8 = 2^3$$
$$9 = 3^2$$
$$25 = 5^2$$

HCF

No common factor.

$$\text{HCF} = 1$$

LCM

$$\text{LCM} = 2^3 \times 3^2 \times 5^2$$
$$\text{LCM} = 1800$$

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✅ Final Answer

(i)

$$\boxed{\text{HCF} = 3,\ \text{LCM} = 420}$$

(ii)

$$\boxed{\text{HCF} = 1,\ \text{LCM} = 11339}$$

(iii)

$$\boxed{\text{HCF} = 1,\ \text{LCM} = 1800}$$

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⭐ Key Insight

• If numbers have no common prime factor, then:

$$\text{HCF} = 1$$

• LCM uses the highest powers of all prime factors.

🧠 Memory Line:

HCF checks common factors, LCM includes all factors