# A Special System of Linear Congruences

## Main Question or Discussion Point

Suppose I have a system of m equations in k unknowns

\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*}

with the restriction that $0 \le x_n < n$. 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?

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Gaussian elimination will work, if all operations are done mod d (I suspect this d should be n).

What if I get a solution where one of the x's is outside its bounds?
Reduce it mod d (or n).

Gaussian elimination will work, if all operations are done mod d (I suspect this d should be n).
No, it should be mod d.

What if I get a solution where one of the x's is outside its bounds?
Reduce it mod d (or n).
I messed up with the restrictions. It should be $0 \le x_n \le i_n$ for some given $i_n$. If I mod an x by d or by $i_n$, 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?

Then, if after reducing mod d, there is a violation of any of the restrictions, we must conclude that there is no solution.

You have a good point there. Thanks for your help.