Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod p), and ask whether this number lies in a given arith- metic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: ife is a prime power and a is a multiple of(and a is a multiple of 4 if= 2), then the density of primes p of K such that the order of (G mod p) is congruent to a moduloe only depends on a through its-adic valuation.