if i^( 5n +1 ) = 1 , n∈Z then show that : n = 3+4p ,p∈Z


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Question Number 104333 by 175mohamed last updated on 20/Jul/20

$i f i^{5 n + 1} = 1, n \in Z$ $t h e n s h o w t h a t : n = 3 + 4 p, p \in Z$
	Answered by OlafThorendsen last updated on 20/Jul/20

	$i^{5 n + 1} = 1 = i^{4 k}$ $\Leftrightarrow 5 n + 1 = 4 k$ $\Leftrightarrow n = - 1 + 4 k - 4 n$ $\Leftrightarrow n = 3 + 4 (k - n - 1)$ $\Leftrightarrow n = 3 + 4 p$
	Answered by mr W last updated on 21/Jul/20

	$i = \cos \frac{π}{2} + i \sin \frac{π}{2}$ $i^{5 n + 1} = \cos \frac{(5 n + 1) π}{2} + i \sin \frac{(5 n + 1) π}{2}$ $1 = \cos 2 k π + i \sin 2 k π$ $\Rightarrow \frac{(5 n + 1) π}{2} = 2 k π$ $\Rightarrow 5 n + 1 = 4 k$ $\Rightarrow n = 4 k - 4 n - 1$ $\Rightarrow n = 4 (k - n - 1) + 3$ $\Rightarrow n = 4 p + 3$
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