Problem:
There are different complex numbers such that . For how many of these is a real number?
Answer Choices:
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B.
C.
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E.
Solution:
The complex numbers such that are the roots of . The factors can have at most 6,6 , and 12 , roots, respectively. Because has 24 distinct roots, the factors do actually have 6,6 , and 12 distinct roots, respectively. The six roots of the first factor satisfy , and the six roots of the second factor satisfy . The twelve roots of the third factor satisfy , so is never real in this case. There are roots such that is real.
OR
The complex values of such that are the 24 th roots of unity. These values can be written in the form , where is an integer between 0 and 23, inclusive. By Euler's Theorem,
This quantity is a real number if and only if , which occurs if and only if is even. There are therefore complex values of such that is real.
The problems on this page are the property of the MAA's American Mathematics Competitions