Problem: Find all positive integers $n > 1$ such that polynomial $x^4 + 3x^3 + x^2 + 6x + 10$ belongs to the ideal generated by polynomial $x^2 + x + 1 $ in $\mathbb{Z} _n[x]$.
Solution: Note that
\[
x^4 + 3x^3 + x^2 + 6x + 10 = \left( x^2 + x + 1 \right) \left( x^2 + 2x -2 \right) + (6x + 12).
\]
So, the polynomial $x^4 + 3x^3 + x^2 + 6x + 10$ belongs to the ideal generated by $x^2 + x + 1 $ in $\mathbb{Z} _n[x]$ if the remainder vanishes in $\mathbb{Z} _n[x]$. Note that
\[
6(x + 2) \equiv 0 \mod 2, 3, 6.
\]
Therefore, the possible values of $n$ are $2,3$ and $6$.