1. **Stating the problem:** Find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of the numbers 12, 21, and 15. 2. **Formula and rules:** - The HCF (or GCD) of numbers is the greatest number that divides all of them without leaving a remainder. - The LCM of numbers is the smallest number that is a multiple of all of them. - Important relation: $$\text{HCF} \times \text{LCM} = \text{Product of the numbers}$$ only if the numbers are two. For more than two numbers, calculate separately. 3. **Find the prime factors:** - 12 = $2^2 \times 3$ - 21 = $3 \times 7$ - 15 = $3 \times 5$ 4. **Find the HCF:** - Common prime factors with the smallest powers. - Only 3 is common in all three. - So, HCF = 3 5. **Find the LCM:** - Take all prime factors with the highest powers. - From 12: $2^2$, from 21: 7, from 15: 5, and common 3. - So, LCM = $2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7$ - Calculate: $4 \times 3 = 12$, $12 \times 5 = 60$, $60 \times 7 = 420$ 6. **Final answer:** - HCF = 3 - LCM = 420 Therefore, the correct option is c) 3, 420.