$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If $d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $ find all possible values of $d$.
Problem
Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade
Tags: number theory, Divisors, greatest common divisor