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 Repeating decimals (Posted on 2006-09-25)
The decimal expansion of 1/271 repeats with a period of length 5:
.003690036900369 ...

However, it is not the smallest number q for which the decimal expansion of 1/q has a repetition length of 5.

Find the smallest q so that the decimal expansion of 1/q has repetition length n for each of {1, 2, ..., 10}

Is there a simple way of finding such a number?

 Submitted by Jer Rating: 4.5000 (2 votes) Solution: (Hide) 1/3 = .333... 1/11 = .090909... 1/27 = .037037037... 1/101 = .009900990099... 1/41 = .024390243902439... 1/7 = .142857142857142857... 1/239 = .004184100418410041841... 1/73 = 013698630136986301369863... 1/81 = .012345679012345679012345679... 1/451 = .02217294900221729490022172949... If you factor 10^n-1 as Charlie did, http://perplexus.info/show.php?pid=5062&cid=34345 it is apparent the smallest q is a factor of this number. The rule is that q should be the smallest factor that is not a factor of one of the previous rep-9's. For example rep length 12 factors as 3*3*3*7*11*13*37*101*9901. Each of these individual factors occurs in a previous row of the chart, as do many combinations of them (such as 3*3*3*7 which would have rep length 6). The smallest comibination that does not occur higher in the chart is 7*101 which is why 707 is the smallest q with rep length 12. See http://mathworld.wolfram.com/DecimalExpansion.html for more about finding such numbers.

 Subject Author Date re(6): General solution Richard 2006-09-27 15:52:05 re(5): General solution Old Original Oskar! 2006-09-27 11:47:07 re(4): General solution Richard 2006-09-26 23:07:48 re(3): General solution Old Original Oskar! 2006-09-26 16:14:41 re(2): General solution Federico Kereki 2006-09-26 11:34:15 re: General solution Jer 2006-09-26 11:17:03 General solution Federico Kereki 2006-09-26 10:58:27 re(3): computer exploration brianjn 2006-09-25 23:36:50 re(2): computer exploration Jer 2006-09-25 11:46:38 re: computer exploration Charlie 2006-09-25 11:35:02 computer exploration Charlie 2006-09-25 11:03:49

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