Divisors in residue classes, constructively

Let $r,s,n$ be integers satisfying $0 \leq r < s < n$, $s \geq n^{\alpha}$, $\alpha > 1/4$, and let $\gcd(r,s)=1$. Lenstra showed that the number of integer divisors of $n$ equivalent to $r \pmod s$ is upper bounded by $O((\alpha-1/4)^{-2})$. We re-examine this problem, showing how to explicitly construct all such divisors, and incidentally improve this bound to $O((\alpha-1/4)^{-3/2})$.