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How to 'undo' mod, or solve equation with a mod.?

  1. Jan 29, 2010 #1
    if we are given something of the form

    where we are given a and b ...how can x be solved for ?

    what if we are given a range for which x has to fall within?
    something like 0 <= lowerbound <= x <= upperbound

    --

    I have thought about this a little while i came up with that
    with k a positive integer.. would generate all the possible solutions

    is this correct?
    can anyone guide me in right direction to solving this?
     
  2. jcsd
  3. Jan 29, 2010 #2
    What about negative numbers? Or 0? For k.

    You're definitely in the right direction.
     
  4. Jan 29, 2010 #3
    Why must x be greater than a?
     
  5. Jan 29, 2010 #4
    no no, i mean some other defined lowerbound and upperbound.

    for example, 0 to 1000000.. otherwise i assume there would be endless solutions.

    ---
    I guess in my case i want to exclude negatives from k (since my boundary is positive), and include 0.

    ---
    I read some more and I found this :
    So i guess I can say go from a=x(mod b) -> x = a(mod b)

    Is this correct?
     
    Last edited by a moderator: May 4, 2017
  6. Jan 30, 2010 #5
    1. a could be greater than b, so even if you want your x to be positive, k > 0 is too strict a condition.

    2. Sure, x = a (mod b ) iff a = x (mod b). Remember that by definition, x = a (mod b) <=> b | x-a.
     
  7. Jan 30, 2010 #6
    x = kb + a where kb + a > -1 is your general solution based upon your condition that x is not negative. Of course k could be any integer including even zero or a negative number for these conditions to be met.
     
    Last edited: Jan 30, 2010
  8. Jan 30, 2010 #7
    the actual problem I am trying to solve is a bit different .. but similar problem i think (?)
    okay, so with these problem settings below

    (U being some integer to represent x's maximum value)

    I can try to solve for x by setting up

    and the solutions are when both x and k are integers.

    ah i understand, since b wasn't specified to fall into any certain range :) , hm since we say now that it is always greater than x,

    (I think I can) assume that k >= 0.

    so is it right to think that there are 1 + FLOOR(b/U) solutions for x?
     
    Last edited: Jan 30, 2010
  9. Jan 31, 2010 #8
    Bear in mind that there could be no solutions at all. For example,
    [tex]x^2 \equiv 2 \pmod 4[/tex]
    has no solutions, since all squares modulo 4 are either 0 or 1.

    You can check this wikipedia entry,
    http://en.wikipedia.org/wiki/Quadratic_residue
     
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