Question 3

We will find the prime factors of each number. Then we use these factors to find the HCF and LCM.

Step 1 — For 12, 15 and 21

Let's find the prime factors of each number.

$12 = 2 \times 2 \times 3$

$12 = 2^2 \times 3^1$

$15 = 3 \times 5$

$15 = 3^1 \times 5^1$

$21 = 3 \times 7$

$21 = 3^1 \times 7^1$

To find the HCF, we take the lowest power of common prime factors. The only common prime factor is 3.

$\boxed{\text{HCF}(12, 15, 21) = 3}$

To find the LCM, we take the highest power of all prime factors. The prime factors are 2, 3, 5, 7.

$\text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1$

$= 4 \times 3 \times 5 \times 7$

$= 12 \times 35$

$\boxed{\text{LCM}(12, 15, 21) = 420}$

Step 2 — For 17, 23 and 29

Let's find the prime factors of each number. These numbers are all prime numbers.

$17 = 17^1$

$23 = 23^1$

$29 = 29^1$

To find the HCF, we take the lowest power of common prime factors. There are no common prime factors other than 1.

$\boxed{\text{HCF}(17, 23, 29) = 1}$

To find the LCM, we take the highest power of all prime factors. The prime factors are 17, 23, 29.

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

$= 391 \times 29$

$\boxed{\text{LCM}(17, 23, 29) = 11339}$

Step 3 — For 8, 9 and 25

Let's find the prime factors of each number.

$8 = 2 \times 2 \times 2$

$8 = 2^3$

$9 = 3 \times 3$

$9 = 3^2$

$25 = 5 \times 5$

$25 = 5^2$

To find the HCF, we take the lowest power of common prime factors. There are no common prime factors other than 1.

$\boxed{\text{HCF}(8, 9, 25) = 1}$

To find the LCM, we take the highest power of all prime factors. The prime factors are 2, 3, 5.

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

$= 8 \times 9 \times 25$

$= 72 \times 25$

$\boxed{\text{LCM}(8, 9, 25) = 1800}$

Answer

(i) HCF = 3, LCM = 420 (ii) HCF = 1, LCM = 11339 (iii) HCF = 1, LCM = 1800