Question:
The least positive integer n such that $\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}, \mathrm{i}=\sqrt{-1}$ is a positive integer, is________
Solution:
$\frac{(2 i)^{n}}{(1-i)^{n-2}}=\frac{(2 i)^{n}}{(-2 i)^{\frac{n-2}{2}}}$
$=\frac{(2 i)^{\frac{n+2}{2}}}{(-1)^{\frac{n-2}{2}}}=\frac{2^{\frac{n+2}{2}} i^{\frac{n+2}{2}}}{(-1)^{\frac{n-2}{2}}}$
This is positive integer for $n=6$