First slide

Algebra of complex numbers

Question

Suppose A is a complex number and $n \in N$ , such that Aⁿ = (A + 1)ⁿ = 1, then the least value of n is

Moderate

A

3
B

6
C

9
D

12

Solution

Let A = x + iy. Given,
$| A | = 1 \Rightarrow x^{2} + y^{2} = 1$
and $| A + 1 | = 1 \Rightarrow (x + 1)^{2} + y^{2} = 1$
$\begin{array}{l} \Rightarrow x = - \frac{1}{2} and y = \pm \frac{\sqrt{3}}{2} \\ \Rightarrow A = ω or ω^{2} \\ \Rightarrow (ω)^{n} = (1 + ω)^{n} = {(- ω^{2})}^{n} \end{array}$
Therefore, n must be even and divisible by 3.

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