  The aim of this work is to show that the moduli space $M_{10}$ introduced by
O'Grady in \cite{OG1} is a $2-$factorial variety. Namely, $M_{10}$ is the
moduli space of semistable sheaves with Mukai vector $v:=(2,0,-2)\in
H^{ev}(X,\mathbb{Z})$ on a projective K3 surface $X$. As a corollary to our
construction, we show that the Donaldson morphism gives a Hodge isometry
between $v^{\perp}$ (sublattice of the Mukai lattice of $X$) and its image in
$H^{2} (\widetilde{M}_{10},\mathbb{Z})$, lattice with respect to the Beauville
form of the $10-$dimensional irreducible symplectic manifold
$\widetilde{M}_{10}$, obtained as symplectic resolution of $M_{10}$. Similar
results are shown for the moduli space $M_{6}$ introduced by O'Grady in
\cite{OG2}.
