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2021 AMC 12B Problem 20

Problem 20 of 25HarderAlgebraNumber Theory

Let Q(z)Q(z) and R(z)R(z) be the unique polynomials such that z2021+1=(z2+z+1)Q(z)+R(z) \begin{aligned} &z^{2021}+1 \\ &\quad = (z^2+z+1)Q(z)+R(z) \end{aligned} and the degree of RR is less than 2.2. What is R(z)?R(z)?

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Solution

Since z3≡1(modz2+z+1)z^3\equiv 1\pmod{z^2+z+1} and 2021=3⋅673+2,2021=3\cdot 673+2, we have z2021≡z2.z^{2021}\equiv z^2. So z2021+1≡z2+1.z^{2021}+1\equiv z^2+1. Reducing further with z2≡−z−1,z^2\equiv -z-1, this is −z−1+1=−z.-z-1+1=-z. Therefore R(z)=−z.R(z)=-z. Thus, the correct answer is A.

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Concepts: polynomial · roots of unity · modular arithmetic

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.