I notice that the problem is asking about differences of distinct members of a set, so I think about the absolute value function, which measures the distance between two numbers on the number line. I also notice that the set is consecutive positive integers from 1 to 16, so the smallest possible difference is 1 and the largest possible difference is 15. I wonder if I can represent every integer from 1 to 15 as a difference of two distinct members of the set. I start with 1, and I see that I can get it as $2 - 1$, $3 - 2$, ..., or $16 - 15$. I move on to 2, and I see that I can get it as $3 - 1$, $4 - 2$, ..., or $16 - 14$. I continue this way, and I notice that for each difference $d$, I can get it as $(d + 1) - 1$, $(d + 2) - 2$, ..., or $16 - (16 - d)$. This means that I can represent every integer from 1 to 15 as a difference of two distinct members of the set, and there are no other possible differences, since 15 is the largest. Therefore, the answer is 15. 
How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, \ldots, 14, 15, 16 \}?$