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  4. The smallest positive integer n for which (1+√3 i)n / 2 is real is

Q. The smallest positive integer $n$ for which $(1+\sqrt{3} i)^{n / 2}$ is real is

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Solution:

$1+ i \sqrt{3}=2 \text{cis} \frac{\pi}{3} $
$ \therefore (1+i \sqrt{3} i)^{n / 2}$
$=2^{\frac{n}{2}}\left(\cos \frac{n \pi}{6}+i \sin \frac{n \pi}{6}\right) $
$\therefore \sin \frac{n \pi}{6}=0 $
$\Rightarrow n=6$ lease value is 6

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