13. Does there exist a positive integer $n$ such that the complex number $z=\left(\frac{3}{\frac{3}{2}+\frac{\sqrt{3}}{2} \mathrm{i}}\right)^{n}$ is a pure imaginary number? If it exists, find the value of the positive integer $n$; if not, explain the reason.
13. Let $w=-\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}$, then $w^{3}=1$. Therefore, $z=\left(\frac{3-\sqrt{3} \mathrm{i}}{2}\right)^{n}=\left[\sqrt{3} \mathrm{i}\left(-\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right)\right]^{n}=(\sqrt{3})^{n} \cdot \mathrm{i}^{n} \cdot w^{n}$, so when $n=3 k\left(k \in \mathbf{N}^{*}\right)$, $z$ is a pure imaginary number.