教室里有一排 35 个衣钩。Paulina 希望衣服之间的间隔相等，使得第一件衣服前、最后一件衣服后以及每两件衣服之间的空衣钩数量相同。假设至少有 1 件衣服和 1 个空衣钩。请问有多少种不同的衣服数量可以满足 Paulina 的要求？

A. 2

B. 4

C. 5

D. 7

E. 9

1. 2025 AMC 8 官方解析

Answer (D): Suppose the coat hooks are numbered from #1 to #35. Imagine an extra hook #36 with a coat on it. If the coats are evenly spaced, then the 36 hooks must be divided into some number of equal-sized blocks, each consisting of a sequence of empty hooks, followed by 1 coat on a hook. For example, if there are 3 blocks with 12 hooks in each block, then coats would hang on hooks #12 and #24 (and the imaginary #36) with 11 empty hooks on either side of each coat. In this way, every factor of 36 corresponds to a possible number of blocks:

Excluding the case with 1 block, which has no coats on the original 35 hooks, and the case with 36 blocks, which leaves no empty hooks, there are 7 different numbers of coats that can satisfy Paulina's pattern, namely 1, 2, 3, 5, 8, 11, and 17.

OR

Let $n$ represent the number of coats. Then there are $35 - n$ empty hooks that must be evenly distributed across the $n + 1$ gaps between the coats, including the ends of the row. Let integer $m$ represent the number of empty hooks in each gap. Then
$m = \frac{35 - n}{n + 1}$
which can be simplified by cross-multiplying and gathering terms to produce
$mn + m = 35 - n$
$mn + m + n = 35$
Adding 1 to both sides and factoring gives
$(m + 1)(n + 1) = 36$
It follows that the possible values of $n$ can be determined by examining $n + 1$, which must be a factor of 36. There are 9 factors of 36. Excluding the factor $n + 1 = 1$, which corresponds to $n = 0$ coats, and the factor $n + 1 = 36$, which corresponds to $n = 35$ coats with no empty hooks, the remaining 7 factors all yield a valid value for $n$. Therefore there are 7 different numbers of coats that can satisfy Paulina's pattern.