# Gomory group

$LaTeX: \left\{v \in \mathbb{Z}^n : \sum_j r(A_j)v_j = r(b),\; v \ge 0 \right\},$
where $LaTeX: v$ corresponds to the nonbasic variables and $LaTeX: r(d)$ is the remainder, $LaTeX: d\!\mod |B|$, where $LaTeX: |B|$ is the determinate of the basis, $LaTeX: B$. The idea is that the nonbasic value chosen in this group, together with the basic levels imputed by the equations $LaTeX: Ax=b$, yields an integer solution (but a basic level might be negative).