}

Find all natural $n>2$, for which the polynomial $x^{n}+x^{2}+1$ is divisible by the polynomial $x^{2}+x+1$.

#
Let $n=3 k+r$, where $0 \leq r \leq 2$. Since $x^{3 k+r}-x^{r}=x^{r}\left(x^{3 k}-1\right)$ is divisible by $x^{3}-1$ and, therefore, by $x^{2}+x+1$, it is sufficient to determine whether the polynomial $x^{r}+x^{2}+1$ is divisible by $x^{2}+x+1$.

For $r=1$ this is true, while for $r=0$ and $r=2$ the corresponding polynomials ($x^{2}+2$ and $2 x^{2}+1$) are not divisible by $x^{2}+x+1$.

## Answer

$n=3 k+1, k \in \mathbf{N}$.