Divisors in residue classes, constructively

Let r, s, n be integers satisfying 0 ≤ r < s < n, s ≥ n, α > 1/4, and gcd(r, s) = 1. Lenstra showed that the number of integer divisors of n equivalent to r (mod s) is upper bounded by O((α − 1/4)). We re-examine this problem; showing how to explicitly construct all such divisors and incidentally improve this bound to O((α−1/4)−3/2).