What is the remainder when $9 + 9^2 + 9^3 + \dots + 9^8$ is divided by 8?

Explanation

1. 01

   $9 = 8 + 1$, so every power $9^k$ leaves remainder 1 on division by 8.

2. 02

   There are 8 terms, so the sum leaves $1 + 1$ + … (8 times) = 8.

3. 03

   8 leaves remainder 0 on division by 8, so the remainder is 0.