SEEMOUS 2010 Problem 4

Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$ . Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as $$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$ where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$ , and $N_i\ne N_j$ for $i\ne j$ .