Question 2

We will use prime factorization to find the LCM and HCF. Then we will verify the product rule.

Step 1 — Find LCM and HCF for 26 and 91

Let's find the prime factors.

$26 = 2 \times 13$

$91 = 7 \times 13$

We find common prime factors. We take the lowest power.

$\boxed{\text{HCF} = 13}$

We find all prime factors. We take the highest power.

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

$\boxed{\text{LCM} = 182}$

Let's multiply the given numbers.

$\text{Product of numbers} = 26 \times 91$

$\boxed{2366}$

Let's multiply the HCF and LCM.

$\text{HCF} \times \text{LCM} = 13 \times 182$

$\boxed{2366}$

Both results are the same. The property is verified.

Step 2 — Find LCM and HCF for 510 and 92

Let's find the prime factors.

$510 = 2 \times 3 \times 5 \times 17$

$92 = 2 \times 2 \times 23 = 2^2 \times 23$

We find common prime factors. We take the lowest power.

$\boxed{\text{HCF} = 2}$

We find all prime factors. We take the highest power.

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

$= 4 \times 3 \times 5 \times 17 \times 23$

$= 12 \times 5 \times 17 \times 23$

$= 60 \times 17 \times 23$

$= 1020 \times 23$

$\boxed{\text{LCM} = 23460}$

Let's multiply the given numbers.

$\text{Product of numbers} = 510 \times 92$

$\boxed{46920}$

Let's multiply the HCF and LCM.

$\text{HCF} \times \text{LCM} = 2 \times 23460$

$\boxed{46920}$

Both results are the same. The property is verified.

Step 3 — Find LCM and HCF for 336 and 54

Let's find the prime factors.

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

$54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3$

We find common prime factors. We take the lowest power.

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

$= 2 \times 3$

$\boxed{\text{HCF} = 6}$

We find all prime factors. We take the highest power.

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

$= 16 \times 27 \times 7$

$= 432 \times 7$

$\boxed{\text{LCM} = 3024}$

Let's multiply the given numbers.

$\text{Product of numbers} = 336 \times 54$

$\boxed{18144}$

Let's multiply the HCF and LCM.

$\text{HCF} \times \text{LCM} = 6 \times 3024$

$\boxed{18144}$

Both results are the same. The property is verified.

Answer

(i) HCF = 13, LCM = 182. Product of numbers = 2366, HCF × LCM = 2366. Verified. (ii) HCF = 2, LCM = 23460. Product of numbers = 46920, HCF × LCM = 46920. Verified. (iii) HCF = 6, LCM = 3024. Product of numbers = 18144, HCF × LCM = 18144. Verified.