2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/171137Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of n_R and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every l in S the l-adic valuation of the order of (R mod p) equals v_l(m).Final version. To appear on Journal of Number TheoryNumber Theory14K15 (Primary) 11G10, 14G25, 14L15, 11R45 (Secondary)text