Two numbers have product 7168 and highest common factor (HCF) 16. Considering all possible such pairs, find the sum of all the distinct numbers involved.

Introduction / Context:
When product and HCF are known, we can parameterize the two numbers as multiples of the HCF with co-prime multipliers. This approach lets us enumerate all valid unordered pairs and then compute the requested sum over distinct values appearing in those pairs.

Given Data / Assumptions:

• a * b = 7168.
• HCF(a, b) = 16.
• a, b are positive integers.

Concept / Approach:
Let a = 16x and b = 16y with HCF(x, y) = 1. Then 256xy = 7168 ⇒ xy = 7168 / 256. Factor the right-hand side, and list the coprime factor pairs (x, y). Each gives a valid unordered pair (a, b). Finally, add all distinct numbers obtained from these pairs.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 860, 460 do not match the computed sum of distinct elements across all valid pairs.
• “Data inadequate” is incorrect because the parameterization produces a finite, explicit set of pairs.

Common Pitfalls:

• Including non-coprime pairs like (2, 14) which would make the HCF greater than 16.

Final Answer: