Question

Show that $9^n+7$ is divisible by $8,$ where n is a positive integer.

Answer 1

$9^n+7={(1+8)}^n+7$
$={}^n{\text{C}}_0+{}^n{\text{C}}_1.8+{}^n{\text{C}}_2{.8}^2+...+{}^n{\text{C}}_n{.8}^n+7$
$=8\cdot C_1+8^2\cdot C_2+...+C_n{8}^n+8$
$=8\lambda ,$ where $\lambda$ is a positive integer
Hence, $9^n+7$ is divisible by $8$