How many complex numbers satisfy the equation z^5=z, where z is the conjugate of the complex number z?

2023 AMC 12A Problem 14

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14.

How many complex numbers satisfy the equation

z^5=\overline{z},

where

\overline{z}

is the conjugate of the complex number

z?

2

3

5

6

7

Take absolute values:

|z|^5=|z|,

|z|=0

|z|=1

Multiplying by

z

gives

z^6=z\overline{z}=|z|^2;

handle

z=0

separately

Taking magnitudes gives

|z|^5=|z|,

|z|=0

|z|=1.

The value

z=0

works, giving one solution.

|z|=1,

multiply the equation by

z

to get

z^6=z\overline{z}=|z|^2=1.

This has

6

distinct roots, all of modulus

1.

Altogether there are

1+6=7

solutions.

Thus, the correct answer is E.