【正确答案】 D

【答案解析】The logical complexities involved in this question are lessened by testing individual values of k.Does k = 1 satisfy the condition? NO. If n = 1, then n/k = 1/1 = 1 is not midway between consecutive integers.Does k = 2 satisfy the condition? YES. For each positive odd integer n, the value of n/k = n/2 is anodd integer divided by 2, and hence is midway between consecutive integers.Does k = 3 satisfy the condition? NO. If n = 1, then n/k = 1/3 is not midway between consecutive integers.It is easy to see that no other positive integer satisfies the condition, by considering the value of n/k when n = 1. Therefore, 2 is the only positive integer value of k that satisfies the condition.The correct answer is D.