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Roots of Unity

Source: 2023 AMC 12A #14

November 9, 2023
AMCAMC 12AMC 12A2023 AMC2023 AMC 12A

Problem Statement

How many complex numbers satisfy the equation z5=z‾z^{5}=\overline{z}, where z‾\overline{z} is the conjugate of the complex number zz?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7<span class='latex-bold'>(A)</span>~2\qquad<span class='latex-bold'>(B)</span>~3\qquad<span class='latex-bold'>(C)</span>~5\qquad<span class='latex-bold'>(D)</span>~6\qquad<span class='latex-bold'>(E)</span>~7
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