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.