Let $\displaystyle M$ be the $\displaystyle \mathbb{Z}[i]$-module generated by the elements $\displaystyle v_1$, $\displaystyle v_2$ such that $\displaystyle (1+i)v_1+(2-i)v_2=0$ and $\displaystyle 3v_1+5iv_2=0$. Find an integer $\displaystyle r \geq 0$ and a torsion $\displaystyle \mathbb{Z}[i]$-module $\displaystyle T$ such that $\displaystyle M \cong \mathbb{Z}[i]^r \times T$.