a,
$ a^6 - 1 = (a^3 - 1)(a^3 + 1) = (a - 1)(a^2 + a + 1)(a + 1)(a^2 - a + 1) \\ a = 7k + 1 \\ \Rightarrow a - 1 = 7k + 1 - 1 = 7k \vdots 7 \\ \Rightarrow a^6 - 1 \vdots 7 \\ a = 7k + 2 \\ \Rightarrow a^2 + a + 1 = (7k + 2)^2 + 7k + 2 + 1 = 49k^2 + 28k + 4 + 7k + 2 + 1 = 49k^2 + 35k + 7 = 7(7k^2...