Vrftsjtryk

I have to write program which is solving linear congruences withe the same modulo. I have system of congruences like that(only 2 unknowns x and y):
$$begin{cases}
a_1x+b_1y equiv c_1pmod n \
a_2x+b_2y equiv c_2pmod n \
end{cases}
$$

I'm using Cramer's rule now and it works when the modulus is prime, but when modulus is not prime, my program behaves incorrectly.

This is not as simple as it may look. For example, there may be multiple solutions for even a single equation ($2xequiv 0 pmod 4$ has 2 solutions$pmod 4$, for example). This breaks the nice theory one knows from the reals. That it works for primes $p$ is due to the fact that the the numbers you are working with also form a field, just like the reals.

It seems you have to go the hard way and

1. factor $n$ into prime powers: $n=p_1^{alpha_1}p_2^{alpha_2}ldots$,


2. you need to solve the problem independently for each prime power $p_i^{alpha_i}$ (see next steps) and at the end combine the results using the Chineese reminder theorem.


3. for each prime power $p_i^{alpha_i}$, first solve it$pmod {p_i}$. This will work with what you have it seems. This gives you for all variables either a result$pmod {p_i}$ or no information (if the coefficinet before the variable was a multiple of $p_i$ in all equations). Introduce new variables ($x_i'$) for the ones you got results for ($x_i$) in the form of $x_i=p_ix_i' + r_i$.


4. reformulate your equations using the new $x_i'$ variables and the older ones that did not yield information in step 3. You will see that now all coefficients before variables contain the factor $p_i$. Now it's time to consider$pmod {p_i^2}$


5. You can divide all your equations by $p_i$. If the constant term is not divisible by $p_i$, the system has no solution. You must also divide the modulus by $p_i$, this is an equivalent operation! ($paequiv pb pmod {p^2}$ means the same as $aequiv b pmod {p}$.


6. Now you are back where you started, you can solve the system with (possibly) some new variables $pmod {p}$ and continue as in steps 3-5. Each time you get information about your variables modulo a higer power of $p_i$, and at the end you get them for what you need, $pmod {p_i^{alpha_i}}$
   

Again, this is probably at least a few days work, not a few hours. Maybe somebody else has a better idea, though.