divisibility - Any digit written $6k$ times forms a number divisible by $13$

Any digit written $6k$ times (like $111111$, $222222222222222222222222$, etc.) forms a number divisible by $13$. (source: a solution taken from careerbless)

I tested with many numbers and it seems this is correct. But, is it possible to prove this mathematically? If so, it will be a convincing statement. Please help. I am not able to think how such properties can be proved.

Answer

Here's an overview of the proof:

1. Prove $111111$ is a multiple of $13$. (Hint: Use a calculator.)


2. Prove that all numbers with a digit written $6k$ times is a multiple of $111111$. You can do this by splitting a number up into groups of $6$ digits like this:
   $$222222222222222222=222222000000000000+222222000000+222222$$