  Let f be an integer greater than one. We study three progressively finer
equivalence relations on closed 3-manifolds generated by Dehn surgery with
denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f
is odd, weak f-congruence preserves the ring structure on cohomology with
Z_f-coefficients. We show that strong f-congruence coincides with a relation
previously studied by Lackenby. Lackenby showed that the quantum SU(2)
invariants are well-behaved under this congruence. We strengthen this result
and extend it to the SO(3) quantum invariants. We also obtain some
corresponding results for the coarser equivalence relations, and for quantum
invariants associated to more general modular categories. We compare S^3, the
Poincare homology sphere, the Brieskorn homology sphere Sigma(2,3,7) and their
mirror images up to strong f-congruence. We distinguish the weak f-congruence
classes of some manifolds with the same Z_f-cohomology ring structure.
