A vector of positive integer numbers with $n$ coordinates is given $a=(a\_1,\ldots,a\_n)$. It holds that $a\_1+\cdots+a\_n$ is divisible by some positive integer number $k$. I have checked many cases and arrived to the conjecture that one can always find at most $n$ vectors with $n$ non-negative integer coordinates such that in all the vectors the sum of the coordinates is exactly equal to $k$ and $a$ is represented as a positive integer combination of these vectors.


Example: $n=3$, $k=5$ and $a = (12,7,6)$, then the $3$ vectors satisfying above described property are $(2,2,1)$, $(5,0,0)$ and $(1,1,3)$, because $a = 3\cdot (2,2,1) + 1\cdot (5, 0, 0) + 1\cdot (1,1,3)$.


One can manually show that the conjecture holds for $k=2,3,4,5$ or $n=2$. It is also easy to see that only $n<k$ is an interesting case, $n\geq k$ can be reduced to the former.



Positive integer combination of non-negative integer vectors
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](https://arxiv.org/abs/math/0501246), 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 $k$.


