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Abstract Algebra: repeating decimals and prime factors

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove if m/n has a repeating decimal expansion of period k, and n has no repeated prime factors, then some prime factor of n divides 10k-1 and no number of the form 10j-1 for 1 ≤ j < k

    2. Relevant equations

    3. The attempt at a solution

    I know that if a decimal expansion d has period 5, then
    d(10N+5-10N) is an integer for some number N representing the number of decimal places before the repeating part of the decimal expansion begins.

    I'm really not sure where to go with this problem though.

    I think it would simplify the problem if I knew the repeating portion of m/n began immediately after the decimal point. But I'm not sure how to prove that either.
  2. jcsd
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