The remainder when  $a^9-b^9$  is divided by $a^2+ab+b^2$ is

${\left(a^3\right)}^3-{\left(b^3\right)}^3÷(a^2+b^2+ab)$

$\Rightarrow (a^3-b^3)(a^6+b^6+{\left(ab\right)}^3)÷(a^2+b^2+ab)$

$=\frac{(a-b)(a^2+ab+b^2)(a^6+b^6+{\left(ab\right)}^3)}{(a^2+b^2+ab)}$

$=(a-b)[a^6+b^6+{\left(ab\right)}^3]$

$\because$ $a^2+b^2+ab$ is factor of $a^9-b^9$,  remainder is zero.