Considered as a polynomial in the $a_i$'s, $V$ is never divisible by p, since the monomial $a_1^{n-1}a_2^{n-2}\cdots a_{n-1}$ always appears with coefficient 1. However, by the magic of Fermat's little theorem, it can be that all of its values are divisible by p, even if the polynomial itself isn't. As Mark points out, this happens if and only if $p< n$.
Considered as a polynomial in the $a_i$'s, $V$ is never divisible by p, since the monomial $a_1^{n-1}a_2^{n-2}\cdots a_{n-1}$ always appears with coefficient 1. However, by the magic of Fermat's little theorem, it can be that all of its values are divisible by p, even if the polynomial itself isn't. As Mark points out, this happens if and only if \$p