A vector of positive integer numbers with $n$ coordinates is given $a=(a_1,...,a_n)$. It holds that $a_1+...+a_n$ 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.

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.

A vector of positive integer numbers with $n$ coordinates is given $a=(a_1,...,a_n)$. It holds that $a_1+...+a_n$ is divisible by some positive integer number $k$. I have checked many cases and arrived to the conjecture that there exist 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.

A vector of positive integer numbers with $n$ coordinates is given $a=(a_1,...,a_n)$. It holds that $a_1+...+a_n$ 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.