Modular Arithmetic Question -- Sum of digits of consecutive numbers.

Question

I'm having trouble understanding the following problem.

Take three consecutive integers with the largest being a multiple of three. e.g. 61, 62, 63.

Find the sum of these numbers... 186.

Add these digits together... 1 + 8 + 6 = 15

Add these digits together... 1 + 5 = 6

No matter what consecutive numbers you start with, you always end with 6. How would I go about writing this as a proof.

Thanks

Answer 1

Let the largest be $3k$.

$$(3k-2)+(3k-1)+3k\equiv6(\mod 9)$$

Since there is only one digit left and the sum $(\mod 9)$ is an invariant, it must be $6$ in the end.