2017 AMC 12A Problem 17

Below is the professionally curated solution for Problem 17 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:roots of unity complex number

Difficulty rating: 1910

17.

There are $24$ different complex numbers $z$ such that $z^{24}=1.$ For how many of these is $z^6$ a real number?

$0$

$4$

$6$

$12$

$24$

Solution:

The $24$ solutions are the $24$ th roots of unity, $z=e^{\pi i k/12}$ for $k=0,1,\ldots,23.$

Then $z^6=e^{\pi i k/2}=\cos\dfrac{k\pi}{2}+i\sin\dfrac{k\pi}{2},$ which is real exactly when $\sin\dfrac{k\pi}{2}=0,$ i.e. when $k$ is even. There are $12$ even values of $k$ in the range.

Thus, the correct answer is D.

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