On the number of points in a lattice polytope

In this article we will show that for every natural $d$ and $n>1$ there exists a natural number $t$ such that for every $d$-dimensional simplicial complex $\mathcal{T}$ with vertices in $\mathbb{Z}^d$, the number of lattice points in the $t^{\mathrm{th}}$ dilate of $\mathcal{T}$ is exactly $\chi(\mathcal{T})$ modulo $n$, where $\chi(\mathcal{T})$ is the Euler characteristic of $\mathcal{T}$.