For all n€ R, $9^{n+1}$ - 8n - 9 is divisible by

$\begin{array}{r}{9}^{\mathrm{n}+1}-8\mathrm{n}-9=9^{\mathrm{n}}\times 9-8\mathrm{n}-9\\ =(1+8{)}^{\mathrm{n}}\times 9-8\mathrm{n}-9\\ =\left({ }^{\mathrm{n}}{\mathrm{C}}_0{+}^{\mathrm{n}}{\mathrm{C}}_{1}8{+}^{\mathrm{n}}{\mathrm{C}}_2{8}^2{+}^{\mathrm{n}}{\mathrm{C}}_3{8}^3+\dots {+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{8}^{\mathrm{n}}\right)9-8\mathrm{n}-9\end{array}$

$\begin{array}{r}=\left(1+8\mathrm{n}{+}^{\mathrm{n}}{\mathrm{C}}_2{8}^2{+}^{\mathrm{n}}{\mathrm{C}}_3{8}^3+\dots {+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{8}^{\mathrm{n}}\right)9-8\mathrm{n}-9\\ =9+72\mathrm{n}+\left({ }^{\mathrm{n}}{\mathrm{C}}_2{8}^2{+}^{\mathrm{n}}{\mathrm{C}}_3{8}^3+\dots {+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{8}^{\mathrm{n}}\right)9-8\mathrm{n}-9\\ =(72\mathrm{n}-8\mathrm{n})+8^2\left({ }^{\mathrm{n}}{\mathrm{C}}_2{+}^{\mathrm{n}}{\mathrm{C}}_{3}8+\dots {+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{8}^{\mathrm{n}-2}\right)9\\ =64\mathrm{n}+64\left({ }^{\mathrm{n}}{\mathrm{C}}_2{+}^{\mathrm{n}}{\mathrm{C}}_{3}8+\dots {+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{8}^{\mathrm{n}-2}\right)9\\ =64\left[\mathrm{n}+\left({ }^{\mathrm{n}}{\mathrm{C}}_2{+}^{\mathrm{n}}{\mathrm{C}}_{3}8+\dots {+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{8}^{\mathrm{n}-2}\right)9\right]\end{array}$

= 64 x Some constant numbers
= Divisible by 64