AIME 1986!

$\frac{n^3+100}{n+10}=\frac{\color{teal}{(n^3+1000)}-900}{\color{teal}{n+10}}=\color{teal}{\frac{n³+1000}{n+10}}-\frac{900}{n+10}=\color{teal}{n²-10n+100}-\frac{900}{n+10}\\ \implies \large{\color{#20A900}{\boxed{n+10| 900}}}\\ \text{The largest positive integer that divides}\; 900\;\text{is}\; 900\;\text{itself.Hence:}\\ \implies n+10=900\implies \color{#3D99F6}{\boxed{n=890}}$

We can apply synthetic division to $\large\frac{n^3+100}{n+10}=n^2-10n+100-\frac{900}{n+10}$ ;

$n+10$ must be a factor of $900 \implies n=890$ . To see that we know that ( $900|890^3+100$ );

Thus, the greatest integer $n$ for which $n+10$ divides $900$ is $\boxed{890}$