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       #Post#: 122--------------------------------------------------
       REPEATING DIGITS
       By: eba95 Date: July 30, 2010, 6:58 am
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       Repeating Digits
       You already know that the decimal
       expansion of a rational number
       eventually repeats or terminates
       (which can be viewed as a repeating
       0).
       But I tell you something that
       perhaps you did not know: if the
       denominator of that rational
       number is not divisible by 3, then
       the repeating part of its decimal
       expansion is an integer divisible by
       nine!
       Example:
       1/7 = .142857142857... has repeating
       part 142857. This is divisible by 9.
       41/55 = .7454545... has repeating
       part 45. This is divisible by 9.
       The Math Behind the Fact:
       This rather curious fact can be
       shown easily. If the rational X is
       purely repeating of period P and
       repeating part R, then
       R = 10P X - X = (10P-1) X = (10P-1) (m/
       n).
       Thus R*n = (10P-1)*m is an integer.
       Since (10P-1) is divisible by 9, if n is
       not divisible by 3, then R must be. If
       you like these fun deductions, you
       may enjoy a course in number
       theory!
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