  Let $K$ be a number field, and let $G$ be a finitely generated subgroup of
$K^\times$. Without relying on the Generalized Riemann Hypothesis we prove an
asymptotic formula for the number of primes $\mathfrak p$ of $K$ such that the
order of $(G\bmod\mathfrak p)$ is divisible by a fixed integer. We also provide
a rational expression for the natural density of this set. Furthermore, we
study the primes $\mathfrak p$ for which the order is $k$-free, and those for
which the order has a prescribed $\ell$-adic valuation for finitely many primes
$\ell$. An additional condition on the Frobenius conjugacy class of $\mathfrak
p$ may be considered. In order to establish these results, we prove an
unconditional version of the Chebotarev density theorem for Kummer extensions
of number fields.
