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Math Help - Repeating Decimals 1/7

  1. #1
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    Repeating Decimals 1/7

    Hello,

    How would I go about explaining why all the fractions of seven consist of the same 6 digits cyclically permuted, and how would I explain why the first three digits and last three digits of the fraction add to 999? I believe it has something to do with Fermat's Little Theorem, but I'm not sure exactly how to explain these phenomenon. Any help is appreciated!
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  2. #2
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    Hi,
    in a fast attemp of solving the problem I noticed the following fact (that might be a rough proof).

    First of all, division (when decimals exist) is based in the last digit a [0...9] with 7.
    if it less than 7 must be done: a0
    ie. [10, 20, 30, 40, 50, 60]

    Next note the following:

    10 mod 7=3 ... 10 floor 7=1
    Next we append to 3 the zero so we can find the remainding term:
    30 mod 7=2 ... 30 floor 7=4
    ...
    20 mod 7=6 ... 20 floor 7=2
    ...
    60 mod 7=4 ... 60 floor 7=8
    ...
    40 mod 7=5 ... 40 floor 7=5
    ...
    50 mod 7=1 ... 50 floor 7=7

    where  floor is the integer part of the division.

    There is a period at modulos of 7 with 10, 20, 30, 40, 50, 60

    if you intefere more with that fact you can make a more elegant proof
    Last edited by gdmath; March 17th 2011 at 09:35 AM.
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  3. #3
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    Note that this can also be done via Primitive Root tecniques.
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