I think it's correct.
If you convert 1 - i and 1 + i to Euler form, you will find that 1 - i has an argument of -pi/4 and 1 + i has an argument of pi/4. Both have a mod of sqrt(2).
We want the real part, meaning Im(z) = 0.
The only way this can be done is if you have multiples of 4, since if you take n = 0, 4, 8... and convert into polar form, the imaginary part (isinx) will be 0.
Typing this on the fly so haven't properly been able to work it out but I think your answer's correct.