2006 AIME II Problem 14

Let S_n be the sum of the reciprocals of the non-zero digits of the integers from 1 to 10^n inclusive. Find the smallest positive integer n for which S_n is an integer.

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.