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Let A,A' be elliptic curves or abelian varieties fully of type GSp definedover a number field K. This includes principally polarized abelian varietieswith geometric endomorphism ring Z and dimension 2 or odd. We compare thenumber of points on the reductions of the two varieties. We prove that A and A'are K-isogenous if the following condition holds for a density-one set ofprimes p of K: the prime numbers dividing #A(k_p) also divide #A'(k_p). Wegeneralize this statement to some extent for products of such varieties. Thisrefines results of Hall and Perucca (2011) and of Ratazzi (2012).

The prime divisors of the number of points on abelian varieties