1. For which natural values of $a$ does the equation $\gcd(4|\cos x|, a|\cos 2x|)=2$ have solutions? Find these solutions.
Answer: $\left[\begin{array}{l}a=4 n-2, n \in N ; x=\pi k, k \in Z \\ a=4 n, n \in N ; x= \pm \frac{\pi}{3}+\pi k, k \in Z\end{array}\right.$