Let $\textrm{A}$ be a digit. If the 7-digit number $353808\textrm{A}$ is divisible by 2, 3, 4, 5, 6, 8, and 9, then what is $\textrm{A}$?
For the number to be divisible by 5, its units digit must be a 5 or a 0. In addition, for the number to be divisible by 2, the units digit must be even. Thus, $\textrm{A}$ must be $\boxed{0}$.

Note that when $\textrm{A}=0$, we also have the following:

* The sum of the digits of the number is 27, so the number is divisible by 3 and 9.   * The number formed by the last two digits is 80, which is a multiple of 4, so the number is divisible by 4.

* The number is divisible by 2 and by 3, so it is divisible by 6.

* The number formed by the last three digits is 080, which is a multiple of 8, so the number is divisible by 8.