<s>[INST] 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. 
 [/INST] Let $a_n=3^{3n}+3^{2n}+3^{n} + 1 = (3^3)^n+(3^2)^n+(3^1)^n + (3^0)^n$.
Since 
$$(x-3^3)(x-3^2)(x-3^1)(x-3^0)= x^4 - 40 x^3 + 390 x^2 - 1080 x + 729$$
we have
$$
a_{n+4} =40 a_{n+3} - 390 a_{n+2} + 1080 a_{n+1} - 729a_{n}
$$
The claim that $a_n$ is a multiple of $4$ for all $n$ follows by induction since it is true for $n=0,1,2,3$.
(The actual coefficients in that recurrence do not matter. It only matters that they are integers.)
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