On divisors in many congruence classes

We establish a result on the number of divisors in many congruence classes which includes a result of Lenstra Jr. that applies to a single residue class. Our result applies to divisors satisfying certain quadratic or cubic equations, given a large enough moduli, which seems to be new. Finally, we leave open a generalization of our work which, if true, would imply Rudin's conjecture on the number of squares in arithmetic progressions.