  In this paper we complete the determination of the index of factoriality of
moduli spaces of semistable sheaves on an abelian or projective K3 surface $S$.
If $v=2w$ is a Mukai vector, $w$ is primitive, $w^{2}=2$ and $H$ is a generic
polarization, let $M_{v}(S,H)$ be the moduli space of $H-$semistable sheaves on
$S$ with Mukai vector $v$. First, we describe in terms of $v$ the pure
weight-two Hodge structure and the Beauville form on the second integral
cohomology of the symplectic resolutions of $M_{v}(S,H)$ (when $S$ is K3) and
of the fiber $K_{v}(S,H)$ of the Albanese map of $M_{v}(S,H)$ (when $S$ is
abelian). Then, if $S$ is K3 we show that $M_{v}(S,H)$ is either locally
factorial or $2-$factorial, and we give an example of both cases. If $S$ is
abelian, we show that $M_{v}(S,H)$ and $K_{v}(S,H)$ are $2-$factorial.
