PyPE solution to PE problem 26

    ProjectEuler.net problem 26
    ===========================

    A unit fraction contains 1 in the numerator. The decimal representation of
    the unit fractions with denominators 2 to 10 are given:

        ^(1)/_(2) = 0.5
        ^(1)/_(3) = 0.(3)
        ^(1)/_(4) = 0.25
        ^(1)/_(5) = 0.2
        ^(1)/_(6) = 0.1(6)
        ^(1)/_(7) = 0.(142857)
        ^(1)/_(8) = 0.125
        ^(1)/_(9) = 0.(1)
        ^(1)/_(10) = 0.1

    Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can
    be seen that ^(1)/_(7) has a 6-digit recurring cycle.

    Find the value of d < 1000 for which ^(1)/_(d) contains the longest
    recurring cycle in its decimal fraction part.

    Solution
    --------

    The aide of Sam May was employed:
    (http://samwiseandthestereotypical.blogspot.com/2008/12/clojure-project-euler-and-recurring.html)

    For each value of d, there are three possible cases:

        1) d is coprime with 10
            For any value of d that is coprime with 10, that is, they share no
            common positive factor other than 1, the period of the recurring
            decimal is given by the multiplicative order k. This is the smallest
            value of k satisfying 10**k % d = 1.

        2) d is a power of 2, 5, or 10
            If d is a power of 2, 5, or 10, the decimal number is terminating.
            We pre-compute a list of powers of 2, 5, and 10 less than or equal
            to the limit value -- any d that is in this list will have a period
            of 0.

        3) d is a multiple of 2 or 5
            If d is a multiple of 2 or 5, the decimal will be made up of a non-
            recurring portion of length s and a recurring portion of period k.
            This can be expressed as 10**(s+k) % d = 10**s. Since we only care
            about the recurring part k, we can simply divide out all of the 2s
            and 5s, like d = 2**a * 5**b * d0, where d0 is some representative
            denominator. The length of k for d will be the same as the length of
            k for d0.

    Answer
    ------

    983