Not really clear about what is being asked. If the $a_i$ are all divisible by the same p (choose one) then this p does divide V. Suppose the $a_i$ are 1 ... n, then if $p < n$ then then with $a_i=1$ there is an $a_j$ with $a_j-a_i=p$. If $p \ge n$ then $p \nmid V$. If $p < n$ then in any n numbers there are two with the same residue mod p (pigeonhole) so $p \mid V$. Is a more general context intended?
If $p < n$ then then with $a_i=1$ there is an $a_j$ with $a_j-a_i=p$. If $p \ge n$ then $p \nmid V$.