If-1-i-3-1-i-3-n-is-an-integer-then-n-is-

Question Number 19434 by Tinkutara last updated on 11/Aug/17

If {(\frac{1 + i \sqrt{3}}{1 - i \sqrt{3}})}^{n} is an integer, then n is

Answered by ajfour last updated on 11/Aug/17

{(\frac{1 + i \sqrt{3}}{1 - i \sqrt{3}})}^{n} = {(\frac{e^{i π / 6}}{e^{- i π / 6}})}^{n} = [x]

\Rightarrow e^{in π / 3} = \pm 1 = e^{ik π}

\Rightarrow n = 3 k where k \in Z .

Commented by Tinkutara last updated on 11/Aug/17

Thank you very much Sir!