Solution
= Solution
Part (ii), with exponent $30$, shows that a prime $q$ fails to divide the sum precisely when $q-1$ divides $30$. The positive divisors are $1,2,3,5,6,10,15,30$. Adding one gives $2,3,4,6,7,11,16,31$, of which the primes are
$$
\boxed{q\in\{2,3,7,11,31\}.}
$$
For these primes the sum is $-1$ modulo $q$; for every other prime it is zero.