8. Given that $n$ is a positive integer, and $7^{n}+2 n$ is divisible by 57, then the minimum value of $n$ is
Solve
$7^{n} \equiv 1(\bmod 3) \Rightarrow 3 \mid 2 n+1 \Rightarrow n=3 k+1$, so $7^{3 k+1}=7 \cdot 343^{k}$ $=7 \cdot(18 \cdot 19+1)^{k} \equiv 7(\bmod 19) \Rightarrow 19|6 k+9 \Rightarrow 19| 2 k+3 \Rightarrow 2 k \equiv-3$ $\equiv 16 \Rightarrow k \equiv 8(\bmod 19) \Rightarrow k_{\min }=8 \Rightarrow n_{\min }=25$.