Interesting Sum

Note $0$ doesn't divide any number except itself (depending on definition), so a number that is divisible by all of its digits cannot have $$ as a digit.

We recall some rules of divisibility for digits (for two-digit numbers):

1. All integers are divisible by $1$ .
2. The second digit is even.
3. The sum of the digits is divisible by 3.
4. If the first digit is even, the second is $4$ , $8$ , or $0$ . If the first digit is odd, then the second is $2$ or $6$ .
5. The second digit is $0$ or $5$ .
6. The digits sum to a multiple of $3$ and the second digit is even.
7. The only two-digit multiples of $7$ with $7$ as a digit are $70$ and $77$ .
8. The only two-digit multiples of $8$ with $8$ as a digit are $48$ , $80$ and $88$ .
9. The only two-digit multiples of $9$ with $9$ as a digit are $90$ and $99$ .

Combining all these rules give 14 two-digit numbers: $11,12,15,22,24,33,36,44,48,55,66,77,88,99.$

The sum of these numbers is $\boxed{630}$ .