Question 27
Determine all integers $n \geqslant 2$ with the following property: for any integers $a_{1}, a_{2}, \ldots, a_{n}$ whose sum is not divisible by $n$, there exists an index $1 \leqslant i \leqslant n$ such that none of the numbers
$$
a_{i}, a_{i}+a_{i+1}, \ldots, a_{i}+a_{i+1}+\cdots+a_{i+n-1}
$$
is divisible by $n$. (We let $a_{i}=a_{i-n}$ when $i\gt n$.)
$$
a_{i}, a_{i}+a_{i+1}, \ldots, a_{i}+a_{i+1}+\cdots+a_{i+n-1}
$$
is divisible by $n$. (We let $a_{i}=a_{i-n}$ when $i\gt n$.)