For any ϵ > 0, there exists q 0 (ϵ) such for any q ≥ q 0 (ϵ) and any invertible residue class a modulo q, there exists a natural number that is congruent to a modulo q and that is the product of exactly three primes, all of which are below q 3 2 +ϵ. If we restrict our attention to odd moduli q that do not have prime factors congruent to 1 mod 4, we can find such primes below q 11 8 +ϵ. If we further restrict our set of moduli to prime q that are such that (q-1, 4 • 7 • 11 • 17 • 23 • 29) = 2, we can find such primes below q 6 5 +ϵ. Finally, for any ϵ > 0, there exists q 0 (ϵ) such that when q ≥ q 0 (ϵ), there exists a natural number that is congruent to a modulo q and that is the product of exactly four primes, all of which are below q(log q) 6 .