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Mathematics Question on Complex Numbers and Quadratic Equations

If ω(≠1)\omega (\ne 1) be a cube root of unity and (1+ω2)n=(1+ω4)n,(1 + \omega^2)^n = (1 + \omega^4)^n, then the least positive value of n is

A

2

B

3

C

5

D

6

Answer

3

Explanation

Solution

Given, (1+ω2)n=(1+ω4)n(1+\omega^2)^n=(1+\omega^4)^n
⇒    (−ω)n=(−ω2)n [∵  w3=1 and 1+ω+ω2=0]\Rightarrow \, \, \, \, (-\omega)^n =(-\omega^2)^n \, [\because \, \, w^3=1 \, and \, 1+\omega+\omega^2=0]
⇒     ωn=1\Rightarrow \, \, \, \, \, \omega^n =1
⇒     \Rightarrow \, \, \, \, \, n = 3 is the least positive value of n.