discrete mathematics - Prove that for all positive integers $n, 9|(11^ n − 2 ^n )$

Prove that for all positive integers $n, 9|(11^n − 2^n )$

So the base case would be

9 * k = (11*1 - 2 * 1)
9 * k = 9

k = 1 so yes

The inductive hypothesis would be the fact that $(11^n-2^n)$ is divisible by $9,$

So I thought then I would have to show that $(11^{(n+1)}-2^{(n+1)})$ is divisible by$9$

11^(n+1) - 2^(n+1)
11^(n) * 11^1 - 2^n * 2^1
(11-2) * (11^n-2^n)

9*(11^n-2^n)

Is this algebraically correct?