The H.C.F. (greatest common divisor) of three numbers is 12, and the numbers are in the ratio 1 : 2 : 3. Find the three numbers.

Introduction / Context:
Numbers in a given ratio with a specified HCF can be constructed by multiplying the ratio terms by a common positive integer equal to the HCF divided by the HCF of the ratio terms. Here the ratio is simple and the HCF is provided directly.

Given Data / Assumptions:

• HCF of the three numbers = 12.
• Numbers are proportional to 1 : 2 : 3.

Concept / Approach:
If numbers are 12*k1, 12*k2, 12*k3 with gcd(k1, k2, k3) = 1, and in the ratio 1:2:3, then (k1, k2, k3) = (1, 2, 3). Therefore the actual numbers are 12*1, 12*2, 12*3. This ensures the overall HCF stays at 12 because the inner triple has no common factor beyond 1.

Step-by-Step Solution:

Verification / Alternative check:
Any common factor larger than 12 would imply a common factor among 1, 2, 3, which is impossible. Hence 12, 24, 36 is consistent and minimal with the given ratio.

Why Other Options Are Wrong:

• 10, 20, 30; 5, 10, 15; 4, 8, 12; 6, 12, 18: These do not all have HCF equal to 12, or they do not adhere to the required HCF and ratio simultaneously.

Common Pitfalls:

• Multiplying the ratio terms by an arbitrary number without ensuring the final HCF is exactly 12.

Final Answer:
12, 24, 36