If ( 1 + i 3 1 - i 3 )^n is an integer, then n is - Vidyadip

MCQ

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

A
$1$
B
$2$
✓
$3$
D
$4$

✓

Answer

Correct option: C.

$3$

c
(c) $\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }} = \left( {\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}} \right)\,\left( {\frac{{1 + i\sqrt 3 }}{{1 + i\sqrt 3 }}} \right) = \frac{{ - 2 + i2\sqrt 3 }}{4}$
$ = \,\frac{{ - 1 + i\sqrt 3 }}{2} = \omega $
${\left( {\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}} \right)^n} = {\omega ^n} = {\omega ^3} = 1 \Rightarrow n = 3$.

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