Is it possible to change the underlying field from Euclidean to $\mathbb{F}_p$?

Question

I have an ILP $Ax=b$ that I know has feasible solution(s), and I would like to find a/all feasible solution(s). The issue is that $A,x,b$ are defined over $\mathbb{F}_p$. More specifically, I need the RHS to be equal to $k\mod p$. I'm trying to solve this ILP in MatLab (due to the presence of the graph class, which makes things convenient in other steps), but I'm open to use any other solver than can be obtained for free as a student.

I'm also curious if there is a way to 'Euclideanize' this ILP? As in instead of solving $Ax=b$ defined over $\mathbb{F}_p$, we instead solve $A'x'=b'$ defined over $\mathbb{R}$.

Just to be clear, I'm willing to accept mathematical solutions, but also MatLab/solver setting changes.

Answer 1

Introduce an integer variable $y$ and replace the RHS with $k+py$.