The least positive integer n such that (2i)^n(1-i)^n-2, i=-1, is a positive integer, is An…

Question

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

Answer: 6

Accepted Answer

$\dfrac{(2i)^n}{(1-i)^{n-2}}=(i-1)^n(-2i)$ after simplification. Testing $n=1,2,4,6$ shows $n=1,2,4$ give non-real or non-positive values, while $n=6$ gives $-2i(i-1)^6=16$, a positive integer. Least such $n$ is 6.

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