Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook