Problem Analysis:
The problem states that B is a positive integer such that the difference between its only two distinct factors is odd. We need to find the value of B.

Key Observations:
To solve this problem, we need to understand the concept of factors and their differences. Let's consider some examples to gain insights:

Example 1:
If B = 6, the factors of 6 are 1, 2, 3, and 6. The difference between its two distinct factors, 3 and 2, is 1 (odd).

Example 2:
If B = 12, the factors of 12 are 1, 2, 3, 4, 6, and 12. The difference between its two distinct factors, 6 and 2, is 4 (even).

From these examples, we can infer that the difference between the two distinct factors of a number is always even, except when the number is a perfect square. In the case of a perfect square, the difference between its two distinct factors is odd.

Solution Approach:
Considering the key observations, we can conclude that B must be a perfect square. Let's evaluate the answer choices accordingly:

a) 2: The factors of 2 are 1 and 2, and their difference is 1 (odd). Therefore, B = 2 satisfies the given condition.

b) 3: The factors of 3 are 1 and 3, and their difference is 2 (even). Therefore, B = 3 does not satisfy the given condition.

c) 5: The factors of 5 are 1 and 5, and their difference is 4 (even). Therefore, B = 5 does not satisfy the given condition.

d) 6: The factors of 6 are 1, 2, 3, and 6, and their difference is 5 (odd). Therefore, B = 6 satisfies the given condition.

e) 8: The factors of 8 are 1, 2, 4, and 8, and their difference is 7 (odd). Therefore, B = 8 satisfies the given condition.

Conclusion:
After evaluating all the answer choices, we find that the value of B that satisfies the given condition is 2. Therefore, the correct answer is option (B).