Solution
= Solution
By (ii), failure of divisibility is equivalent to $q-1\mid30$. The positive divisors of $30$ are $1,2,3,5,6,10,15,30$; adding one gives $2,3,4,6,7,11,16,31$. Retaining only <primes> yields
$$
\boxed{q\in\{2,3,7,11,31\}.}
$$
= Solution
By (ii), failure of divisibility is equivalent to $q-1\mid30$. The positive divisors of $30$ are $1,2,3,5,6,10,15,30$; adding one gives $2,3,4,6,7,11,16,31$. Retaining only <primes> yields
$$
\boxed{q\in\{2,3,7,11,31\}.}
$$