Question 67
Find all positive integers $n \lt 2027$ that satisfy the following conditions:
(i) For every positive divisor $d$ of $n$, numbers $1^d, 2^d, \dots, 2026^d$ all have distinct remainders when divided by $2027$.
(ii) $\tau(n)^2 \mid n$ with $\tau(n)$ is the number of positive divisors of $n$.
(i) For every positive divisor $d$ of $n$, numbers $1^d, 2^d, \dots, 2026^d$ all have distinct remainders when divided by $2027$.
(ii) $\tau(n)^2 \mid n$ with $\tau(n)$ is the number of positive divisors of $n$.