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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 Lam and Postnikov, 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 $k$.