What is the least positive integer n for which ((1+i)/(1-i) )^n=1 ?

Correct answer: (D) 4. ((1+i)/(1-i) )^n=1 or [((1+i)(1+i))/((1-i^2 )) ]^n=1 i^n=1 ⇒ n=4

Q162 / 270 · NDA 2010 (Phase 1) · Mathematics

What is the least positive integer $n$ for which $\left(\frac{1+i}{1-i}\right)^n=1$ ?

A
16
B
12
C
8
Correct answer
D
4

Solution

$\begin{aligned} & \left(\frac{1+i}{1-i}\right)^n=1 \\ & \text {or }\left[\frac{(1+i)(1+i)}{\left(1-i^2\right)}\right]^n=1 \\ & i^n=1 \Rightarrow n=4\end{aligned}$

Hence, the correct answer is option: 'D'.

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