2011 AIME II Problem 14

There are N permutations (a_1, a_2, \ldots , a_{30}) of 1, 2, \ldots , 30 such that for m \in \{2,3,5\}, m divides a_{n+m} - a_n for all integers n with 1 \le n < n+m \le 30. Find the remainder when N is divided by 1000.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.