prove that ${3^{3n}} + 3^{2n} + 3^{n } + 1$ is divided by $4$. by induction I tried to take the 3 out but it is not helping me much. 

We have
$$
3^0 \equiv +1 \bmod 4
\\
3^1 \equiv -1 \bmod 4
\\
3^2 \equiv +1 \bmod 4
\\
3^3 \equiv -1 \bmod 4
\\
$$
Therefore, $3^{3n}+3^{2n}+3^{n} + 1 \equiv 1^n +(-1)^n + 1^n + (-1)^n \equiv 0 \bmod 4$.
