<s>[INST] The polynomial $x^{2k}+1+(x+1)^{2k}$ is not divisible by $x^2+x+1$. Find the value of $k \in \mathbb N$. 
The polynomial  $x^{2k}+1+(x+1)^{2k}$ is not divisible by $x^2+x+1$. Find the value of $k\in \mathbb{N}$.

I tried finding out the roots of $x^2+x+1$ which were $\dfrac{-1±\sqrt{3}i}{2}$ but in vain. I got no result other than making the polynomial more complicated.
Here's what I got : $$\left(\frac{-1±\sqrt{3}i}{2}\right)^{2k}+1+\left(\frac{-1±\sqrt{3}i}{2}+1\right)^{2k}.$$
Now, I don't know what to do next.
Any help would be appreciated.
 [/INST] Using the remainder theoram, putting $x^2 + x + 1 = 0$ in the polynomial $P(x) =x^{2k} + 1 + (x + 1)^{2k}$ we will get the remaimder.
$$
P(x) = 1 + x^{2k} + (x^2 + 2x + 1)^k = 1 + x^{2k} + x^k
$$
We can also prove that if $x^2 + x + 1 = 0$, $x^3 = 1$.
We can now clearly see that for $k$ is not equal to $3q$, $P(x)=0$
But given it should not be divisible by $x^2 + x + 1$, Hence $k$ must be $3q$.
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