Verify LCM × HCF = Product of Two Numbers

 

❓ Question

Find the LCM and HCF of the following pairs of integers and verify that:

$$\text{LCM}\times \text{HCF}=\text{Product of the two numbers}$$

(i) 26 and 91
(ii) 510 and 92
(iii) 336 and 54

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🖼️ Solution Image

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✍️ Short Solution

This is a prime factorisation + verification question.

Use:

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

Then verify:

$$\text{LCM} \times \text{HCF} = a \times b$$

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🔹 (i) 26 and 91

Prime Factorisation

$$26 = 2 \times 13$$
$$91 = 7 \times 13$$

HCF

Common factor = 13

$$\text{HCF} = 13$$

LCM

$$\text{LCM} = 2 \times 7 \times 13 = 182$$

Verification

$$13 \times 182 = 2366$$
$$26 \times 91 = 2366$$

✅ Verified

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🔹 (ii) 510 and 92

Prime Factorisation

$$510 = 2 \times 3 \times 5 \times 17$$
$$92 = 2^2 \times 23$$

HCF

Common factor = 2

$$\text{HCF} = 2$$

LCM

$$\text{LCM} = 2^2 \times 3 \times 5 \times 17 \times 23$$
$$\text{LCM} = 23460$$

Verification

$$2 \times 23460 = 46920$$
$$510 \times 92 = 46920$$

✅ Verified

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🔹 (iii) 336 and 54

Prime Factorisation

$$336 = 2^4 \times 3 \times 7$$
$$54 = 2 \times 3^3$$

HCF

$$\text{HCF} = 2^1 \times 3^1 = 6$$

LCM

$$\text{LCM} = 2^4 \times 3^3 \times 7$$
$$\text{LCM} = 3024$$

Verification

$$6 \times 3024 = 18144$$
$$336 \times 54 = 18144$$

✅ Verified

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✅ Final Answer

(i)

$$\boxed{\text{HCF} = 13,\ \text{LCM} = 182}$$

(ii)

$$\boxed{\text{HCF} = 2,\ \text{LCM} = 23460}$$

(iii)

$$\boxed{\text{HCF} = 6,\ \text{LCM} = 3024}$$

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⭐ Key Insight

For two numbers:

$$\boxed{\text{LCM} \times \text{HCF} = \text{Product of the numbers}}$$

🧠 Memory Line:

HCF → lowest common powers, LCM → highest powers