Which one of the following is the minimum possible value of the sum of two positive integers whose product is 24?

Introduction / Context:
This aptitude question tests understanding of factor pairs of a positive integer and how the sum of two numbers behaves when their product is fixed. Such questions are common in problems on numbers and optimisation in basic algebra.

Given Data / Assumptions:
- Two positive integers have a product equal to 24.
- We need the smallest possible value of their sum a + b.
- Only positive integers are considered, because the question explicitly asks for positive integers in the context of basic factor pairs.

Concept / Approach:
When the product a * b is fixed and positive, the sum a + b is smallest when the two positive integers are as close to each other as possible. Therefore, to minimise the sum, we should look for factor pairs of 24 that are nearest to each other in value.

Step-by-Step Solution:
List factor pairs of 24 in positive integers: (1, 24), (2, 12), (3, 8), (4, 6). Compute sums: 1 + 24 = 25, 2 + 12 = 14, 3 + 8 = 11, 4 + 6 = 10. Compare all sums: 25, 14, 11, and 10. The smallest among these sums is 10, which comes from the pair (4, 6). Therefore, the minimum possible value of the sum of the two positive integers is 10.

Verification / Alternative Check:
We can reason that for a constant positive product, making the numbers closer reduces their sum. The pair (4, 6) is the closest pair of positive factors of 24, so its sum should be minimal. Quick recomputation confirms that no other positive factor pair of 24 gives a sum smaller than 10.

Why Other Options Are Wrong:
25 corresponds to the pair (1, 24) and is much larger than 10.
11 corresponds to the pair (3, 8) and is still greater than 10.
8 is not the sum of any positive integer factor pair of 24, so it is invalid in this context.
14 corresponds to the pair (2, 12) and is also larger than 10.

Common Pitfalls:
A common mistake is to ignore the word positive and consider negative factor pairs such as (-4, -6), which also multiply to 24 but give negative sums. In many school level aptitude questions, unless otherwise stated, factor pairs are taken as positive integers. Another error is to stop after checking only one or two factor pairs without systematically listing all possibilities.

Final Answer:
The minimum possible value of the sum of two positive integers whose product is 24 is 10.