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Repeating decimals (Posted on 2006-09-25) Difficulty: 3 of 5
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(6): General solutionRichard2006-09-27 15:52:05
re(5): General solutionOld Original Oskar!2006-09-27 11:47:07
re(4): General solutionRichard2006-09-26 23:07:48
Some Thoughtsre(3): General solutionOld Original Oskar!2006-09-26 16:14:41
Some Thoughtsre(2): General solutionFederico Kereki2006-09-26 11:34:15
re: General solutionJer2006-09-26 11:17:03
SolutionGeneral solutionFederico Kereki2006-09-26 10:58:27
re(3): computer explorationbrianjn2006-09-25 23:36:50
re(2): computer explorationJer2006-09-25 11:46:38
Hints/Tipsre: computer explorationCharlie2006-09-25 11:35:02
Hints/Tipscomputer explorationCharlie2006-09-25 11:03:49
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