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Modular arithmetic with a variable modulus and fractions

  1. Nov 17, 2012 #1
    (This is my first post.)

    I can't seem to find a good way of solving this sort of congruence for x:

    x^2 / 3 + 11 [itex]\equiv[/itex] 5 (mod x)

    Through trial and error it appears at least 3 and 6 are answers, but how can you reach them regularly? (I'm heard conflicting things about fractions being defined for modular arithmetic. It might be that this isn't even a createable congruence.)
     
  2. jcsd
  3. Nov 17, 2012 #2

    haruspex

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    If the denominator is prime to the base then it's always defined. Otherwise, only when the HCF of denominator and base happens to divide the numerator in the ordinary way:
    1/3 (6) does not exist because there is no number x s.t. 3x[itex]\equiv[/itex]1 (6).
    6/15 (9) is ok because HCF(15,9) = 3, which cancels to produce 2/5 (9) = 4.
    Just multiply it out and see what you get.
     
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