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Problem 319

AMC 12 late, AIME early

Number theory Difficulty 4.6 Find the answer

1. For which natural values of $a$ does the equation $\gcd(4|\cos x|, a|\cos 3x|)=2$ have solutions? Find these solutions.

Your answer

Official solution

$\left[\begin{array}{l} a=4 n-2, n \in N ; x=\frac{\pi k}{3}, k \in Z \\ a=4 n, n \in N ; x= \pm \frac{\pi}{3}+\pi k, k \in Z \end{array}\right.$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.