find the smallest positive integer value of n for which {(1+i)^n}/{(1-i)^(n-2)} is a real number ?

((1+i)/(1-i)) = (1+i)^2/2

so ((1+i)/(1-i))^n = (1+i)^(2n)/2^n)

so {(1+i)^n}/{(1-i)^(n-2)} = (1+i)^(2n)/(2^n) (1-i)^2
= ((1+i)^n(1-i))^2 /(2^n)
= (1+i)^(n-1)/2^(n-1)

so (1+i)^(n-1) must be real or n-1 = 0 (by inspection) or n = 1 is the lowest n

if one needs to solve for all n the 1+ i makes 45 degrees with x axis and n = 4t is the solution ( t integer)