2017 AMC 12A Problem 17

Attempt Problem 17 of the 2017 AMC 12A below, then check your answer against the professionally curated solution 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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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

Write

z=e^{\frac{\pi i k}{12}}

for

k=0,1,\ldots,23

Then

z^6=e^{\frac{\pi i k}{2}},

which is real exactly when

k

is even

The

24

solutions are the

24

th roots of unity,

z=e^{\frac{\pi i k}{12}}

for

k=0,1,\ldots,23.

Then

z^6=e^{\frac{\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.

Problem 17 in other years