Numbers from HCF and ratio: The HCF of three numbers is 23. If they are in the ratio 1 : 2 : 3, find the actual numbers.

Introduction / Context:
When numbers are in a given ratio and share a known HCF, each number can be expressed as HCF times the respective ratio part (provided the ratio parts are coprime in aggregate). This is a standard reconstruction technique in HCF–ratio problems.

Given Data / Assumptions:

• HCF = 23
• Ratio = 1 : 2 : 3

Concept / Approach:
Let the numbers be 23x, 23y, 23z with x : y : z = 1 : 2 : 3. Because the common factor extracted is the HCF, the simplest choice is x = 1, y = 2, z = 3 to obtain the minimal triple satisfying the ratio and HCF conditions.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• All other triplets either do not have HCF equal to 23 or do not preserve the exact 1 : 2 : 3 ratio.

Common Pitfalls:

• Multiplying the ratio parts by another common factor, which would increase the HCF beyond 23.

Final Answer: