This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
What 6 digit number, with 6 different digits, when multiplied by all integers
up to 6, circulates its digits through all 6 possible positions, as follows:
ABCDEF * 1 = ABCDEF
ABCDEF * 3 = BCDEFA
ABCDEF * 2 = CDEFAB
ABCDEF * 6 = DEFABC
ABCDEF * 4 = EFABCD
ABCDEF * 5 = FABCDE
ABCDEF=142857 (the digits of the expansion of 1/7).