Suppose `A` is a complex number and `n in N ,` such that `A^n=(A+1)^n=1,` then the least value of `n` is `3` b. `6` c. `9` d. `12`

Options

A. 3
B. 6
C. 9
D. 12

Answer

B

Explanation

Let `A=x+iy.`Given,
`|A|=1rArrx^(2)+y^(2)=1`
and `|A+1|=1rArr(x+1)^(2)+y^(2)=1`
`rArr x=-(1)/(2)andy=pm(sqrt3)/(2)`
`rArr A=omegaoromega^(2)`
`rArr (omega)^(n)=(1+omega)^(n)=(-omega^(2))^(2)`
Therefore, n must be even and divisible by 3.