Question

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

Answer 1

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