Question

The least positive integer $n$ such that $\frac{(2i{)}^n}{(1-i{)}^{n-2}},i=\sqrt{-1}$ is a positive integer, is ___

Answer 1

Answer: 6

$\frac{(2i{)}^n}{(1-i{)}^{n-2}}=\frac{(2i{)}^n}{(-2i{)}^{\frac{n-2}{2}}}$
$=\frac{(2i{)}^{\frac{n+2}{2}}}{(-1{)}^{\frac{n-2}{2}}}=\frac{2^{\frac{n+2}{2};}{i}^{\frac{n+2}{2}}}{(-1{)}^{\frac{n-2}{2}}}$
This is positive integer for $n=6$