Multiples with Only the Digits 0 and 1
Any positive integer number has a multiple containing only the digits 0 or 1. Moreover, if the number is not a multiple of 2 or 5, you can even find a multiple made only with 1s. This Demonstration shows this multiple and additionally, if the number is a power of 2 (2, 4, 8, 16, …), another multiple containing only 1s and 2s.
P. Winkler, Mathematical Puzzles, A Connoisseur's Collection, Natick, MA: AK Peters, 2004.