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A Special System of Linear Congruences

  1. Mar 21, 2010 #1
    Suppose I have a system of m equations in k unknowns

    [tex]
    \begin{align*}
    a_{11}x_1 + a_{12}x_2 + \cdots + a_{1k}x_k &= 0 \pmod d \\
    &\vdots \\
    a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mk}x_k &= 0 \pmod d
    \end{align*}
    [/tex]

    with the restriction that [itex]0 \le x_n < n[/itex]. How do I solve such a thing? Will Gaussian elimination work? What if I get a solution where one of the x's is outside its bounds?
     
  2. jcsd
  3. Mar 23, 2010 #2
    Gaussian elimination will work, if all operations are done mod d (I suspect this d should be n).

    Reduce it mod d (or n).
     
  4. Mar 23, 2010 #3
    No, it should be mod d.

    I messed up with the restrictions. It should be [itex]0 \le x_n \le i_n[/itex] for some given [itex]i_n[/itex]. If I mod an x by d or by [itex]i_n[/itex], it will screw up the solution because when I plug a mod'ed x back into the system, the numbers won't come out right. I would have to mod all the x's by d, by this will automatically happen if I do Gaussian elimination by mod d. However, the problem remains: What if one of the x's is outside its bound?
     
  5. Mar 23, 2010 #4
    Then, if after reducing mod d, there is a violation of any of the restrictions, we must conclude that there is no solution.
     
  6. Mar 23, 2010 #5
    You have a good point there. Thanks for your help.
     
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