



 
Your cyclic division is a lovely observation! Because deep philosophical questions are not my speciality, I’ll not comment on your chirality question except to observe that in Arabic texts, numbers are written in an order (with the units place on the left) that is the reverse of the order we use (with the units place on the right). Perhaps incompatible chirality explains today’s religious problems. The cyclic division you observe is an immediate consequence of our decimal number system. The whole story can be found on the web under the heading “repeated decimals.” (You should also check out the interesting side topic “Midy’s theorem”) I’ve attached a worksheet that I discussed at a math workshop for students in grades 7 to 12. The worksheet was based on a more detailed discussion by Kenneth A. Ross: “Repeating decimals: a period piece". Mathematics Magazine 83:1 (Feb. 2010), 33–45. Here is a very brief version of it using your number 142857. Note that Of course 999,999 is divisible by 37 (and also by 27, 7, 11, and 13), and the same will hold with n/7 replacing 1/7 (which cyclically permutes the digits of your number). Chris Vir wrote back
As far as I can tell, clockwise and anticlockwise are not well defined — write your number clockwise on a transparent page and turn the page over; suddenly your number is anticlockwise! Let’s look at an example: Cyclic permutations of the number 648351 have 27 as their common divisor (which is almost meaningless because ALL permutations of these six digits are divisible by 9 because they sum to a multiple of 9). But cyclic permutations of the reverse number 153846 have 76923 (= 27*7*407) as their common divisor. (That 6digit number comes from multiples of 2/13.) Chris  


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