Let $n_1, n_2, \ldots, n_k$ be positive integers with $\gcd(n_1, n_2, \ldots, n_k) = 1$. Show that every sufficiently large positive integer $n$ can be represented as $n = \sum_i c_i n_i$ with $c_i \geq 0$ integers.
Since $\gcd(n_1,\ldots,n_k)=1$, there exist $a_i$ such that $\sum_i a_i n_i = 1$, so $n = \sum_i (a_i n) n_i$. If $\sum_i b_i n_i = 0$ then $n = \sum_i (a_i n + b_i) n_i$ is another solution. By choosing appropriate $(b_i)$ we can make all $a_i n + b_i \geq 0$ for any $n > (s - n_i - k + 1)n_i$ where $s = \sum_i n_i$. By finding two consecutive integers in the set of representable numbers (using the gcd reduction argument), every sufficiently large integer is representable.
Let $n_1, n_2, \ldots, n_k$ be positive integers with $\gcd(n_1, n_2, \ldots, n_k) = 1$. Show that every sufficiently large positive integer $n$ can be represented as $n = \sum_i c_i n_i$ with $c_i \geq 0$ integers.