  Let $A$ be the product of an abelian variety and a torus over a number field
$K$, and let $m$ be a positive integer. If $\alpha \in A(K)$ is a point of
infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that
the reduction $(\alpha \bmod \mathfrak p)$ is well defined and has order
coprime to $m$. This set admits a natural density, which we are able to express
as a finite sum of products of $\ell$-adic integrals, where $\ell$ varies in
the set of prime divisors of $m$. We deduce that the density is a rational
number, whose denominator is bounded (up to powers of $m$) in a very strong
sense. This extends the results of the paper "Reductions of points on algebraic
groups" by Davide Lombardo and the second author, where the case $m$ prime is
established.
