AIME II 2010 #1

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] be the greatest integer multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when [unparseable or potentially dangerous latex formula] is divided by 1000.

[sqrtneg1]

Since [unparseable or potentially dangerous latex formula], we know that [unparseable or potentially dangerous latex formula]. Therefore, the sum of the digits of [unparseable or potentially dangerous latex formula] must be divisible by [unparseable or potentially dangerous latex formula]. Since [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula] can only consist of the digits [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], and [unparseable or potentially dangerous latex formula]. Thus, the maximum number which can be made under the given conditions is [unparseable or potentially dangerous latex formula], and so the answer is [unparseable or potentially dangerous latex formula].