Question 77
Determine all integers $n \geqslant 2$ verifying the following property: For all integers $a_{1}, a_{2}, \ldots, a_{n}$ whose sum is not divisible by $n$, there is an index $i$ such that none of the numbers $$
a_{i}, a_{i}+a_{i+1}, \ldots, a_{i}+\cdots+a_{i+n-1}
$$ is divisible by $n$ (for $i\gt n$, we set $a_{i}=a_{i-n}$ ).
a_{i}, a_{i}+a_{i+1}, \ldots, a_{i}+\cdots+a_{i+n-1}
$$ is divisible by $n$ (for $i\gt n$, we set $a_{i}=a_{i-n}$ ).