Consider the regular simplex $x_1+x_2+\cdots+x_n=k$ and $x_i\geq 0$. The collection of hyperplanes $x_i=j$ where $1\le i\le n$, $0\le j\le k$, partition our simplex into smaller copies of itself with disjoint interiors. This also partitions the positive orthant $\mathbb R_+^n$ into several cones with disjoint interior.
If you intersect each cone with the set of lattice points whose coordinates sum to a multiple of $k$ you can show that you get a cone isomorphic to $\mathbb N^n$. This shows that each nonnegative vector with coordinates adding to a multiple of $k$ can be written as a nonnegative combination of the $n$ generators of the cone it lies in.
Hint for the proof: The rays of the cone are given by $(a_1,a_2,\dots,a_n)+e_i$ where $1\le i\le n$. Where the $a$'s sum to $-1\pmod{k}$ and the $e_i$'s are the coordinate vectors.