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Given a multiplicative group of nonzero rational numbers and a positive integer m, we consider the problem of determining the density of the set of primes p for which the order of the reduction modulo p of the group is divisible by m. In the case when the group is finitely generated the density is explicitly computed. Some examples of groups with infinite rank are considered.doi:10.1090/s0025-5718-2014-02872-x fatcat:eoqq7pfx3fg4fp5qm5xrws2iea