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Product of products is a root of unity (BMT 2019 Algebra #9)

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May 6, 2019

Problem Statement

Let an a_n be the product of the complex roots of x2n=1 x^{2n} = 1 that are in the first quadrant of the complex plane. That is, roots of the form a+bi a + bi where a,b>0 a, b > 0 . Let r=a1⋯a2⋅…⋅a10 r = a_1 \cdots a_2 \cdot \ldots \cdot a_{10} . Find the smallest integer k k such that r r is a root of xk=1 x^k = 1 .
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