9. The smallest positive integer $n$ that satisfies $I=\left(\frac{1}{2}+\frac{1}{2 \sqrt{3}} \mathrm{i}\right)^{n}$ being a pure imaginary number is $n=$ . $\qquad$
9.3.
$$
\begin{array}{l}
\text { Given } I=\left(\frac{1}{2}+\frac{1}{2 \sqrt{3}} \mathrm{i}\right)^{n} \\
=\left(\frac{1}{\sqrt{3}}\right)^{n}\left(\cos \frac{n \pi}{6}+\mathrm{i} \sin \frac{n \pi}{6}\right)
\end{array}
$$

is a pure imaginary number, we know the smallest value of $n$ is 3.