Given a matrix multiplication and a vector addition. (A,b has rational entries)
$$Ax+b$$
how do i get an $x$ for that $Ax+b$ is integer or show that there is not such a solution? $x$ has no restrictions.
EDIT I am currently trying to find solutions for a reduced problem:
$$Ax$$
you can then set x_i to the common denominator of the column a_i and has a solution.
a) Can you calculate all possible solutions?
b) If you demand A*x >= 0 is there still possible to construct one solution?