Question

If ${\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)}^n$ is an integer, then $n$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$z=1+i\sqrt{3}$

$z^2=-2(1-i\sqrt{3})=-2\overline{z}\Rightarrow {\overline{z}}^2=-2z$

$z^3=-2z\overline{z}=-2\left(4\right)=-8={\overline{z}}^3$

$\Rightarrow n=3$

So, Option C is the correct answer.