A ticket is considered lucky if the sum of the first 3 digits equaled the sum of the last 3 digits.

Question

Show that the sum of these tickets between 000000 and 999999 are divisible by 7.
I started by thinking that I would need to find an interval where the lucky tickets would sum to a multiple of seven and then use that interval to find all of the interval sums between 000000 and 999999, but I have no efficient way of actually finding the numbers. What is this efficient method?

Answer 1

Most of the lucky numbers pair up. For example, $127406$ is lucky; so is $406127$. Their sum is $$ (127 + 406)\times (7 \times 11 \times 13). $$

The ones that don't pair up, like $123123$ are themselves divisible by $7$.