Question 23
Determine the largest natural number $n$ for which the following statement holds:
There exist $n$ distinct positive integers $x_{1}, x_{2}, \ldots, x_{n}$ such that, for every choice of numbers $a_{1}, a_{2}, \ldots, a_{n} \in\{-1,0,1\}$ not all zero, $n^{3}$ does not divide $a_{1} x_{1}+a_{2} x_{2}+\ldots+a_{n} x_{n}$.
There exist $n$ distinct positive integers $x_{1}, x_{2}, \ldots, x_{n}$ such that, for every choice of numbers $a_{1}, a_{2}, \ldots, a_{n} \in\{-1,0,1\}$ not all zero, $n^{3}$ does not divide $a_{1} x_{1}+a_{2} x_{2}+\ldots+a_{n} x_{n}$.