Usually, when we solve a congruence system, we try to find the general solution that fits the first equation and the second equation and the third, and so on.
How should we solve a congruence system such that we find the general solution that fits the first equation or the second equation or the third, and so on?
I.e. we have:
$\left \{\begin{matrix} & x \equiv a_1 \pmod {m_1} & \\ & x \equiv a_2 \pmod {m_2} & \\ & \vdots & \\ & x \equiv a_n \pmod {m_n} & \end{matrix}\right.$
And we want to find $x$ such that at least one of the equations is true.
E.g.
$\left \{\begin{matrix} & x \equiv 1 \pmod {4} & \\ & x \equiv 3 \pmod {4} & \end{matrix}\right.$
Should yield the solution $x \equiv 1 \pmod 2$
$\left \{\begin{matrix} & x \equiv 1 \pmod {2} & \\ & x \equiv 0 \pmod {3} & \end{matrix}\right.$
Should yield $x \equiv 0,1,2,3,5 \pmod 6$
