Consider the regular (n-1)-simplex $x_1+x_2+\cdots+x_n=k$ and $x_i\geq 0$. The collection of hyperplanes $x_i=p$ where $1\le i\le n$, $p\in \mathbb Z$, partition our simplex into smaller polytopes with disjoint interiors. These polytopes are alcoved polytopes in the sense of <a href="https://arxiv.org/abs/math/0501246">Lam and Postnikov</a>, and therefore have unimodular triangulations. By taking the cones over these triangulations you get a unimodular decomposition of $\mathbb N^n$ intersected with your lattice of vectors whose coordinates add to a multiple of $k$.