I like 5's!

It's simple. Write n 5s. Now take the value of n on testing the divisibility of 9&11. The answer is 18. It's the answer for all digits except for 9.

An easy way to solve this involves understanding why the "divisibility test" for dividing by 9 works; then one can realize that a similar test for "divisibility by 99" also works:

Rule: A number is divisible by 99 if and only if the sum of their "base 100" digits is.

A digit in base 100 is just a two-digit number in base 10. So for example $5555 = (55)(55)_{100}$ Now the sum of these "digits" is just a multiple of 55. Then you just need to calculate $lcm(55,99) = 9 \times 55$ which corresponds to the sum of a 9-digit number to base 100, or 18-digit number to base 10.