Problem 1162

1. Identify the digits that cannot be in the number:
- Digit 0: If 0 were a digit of $n$, then $n$ would not be a multiple of 0 (since division by zero is undefined). Therefore, 0 cannot be a digit of $n$.
- Digit 5: If 5 were a digit of $n$, then $n$ must end in 5 to be divisible by 5 (since 0 is not a digit). This would make $n$ odd, and thus $n$ cannot have any even digits, which is impossible since we need 7 different digits. Therefore, 5 cannot be a digit of $n$.
- Digit 4: If 4 were a digit of $n$, then we need to check the sum of the digits to ensure it is a multiple of 3 and 9. Removing any one digit from $\{1,2,3,6,7,8,9\}$ and adding 4 does not result in a sum that is a multiple of 9. Therefore, 4 cannot be a digit of $n$.

2. Determine the remaining digits:
- The remaining digits are $\{1, 2, 3, 6, 7, 8, 9\}$.

3. Verify the construction:
- A valid construction of $n$ using these digits is $n = 3678192$.
- Check if $n$ is divisible by each of its digits:
- $3678192 \div 1 = 3678192$ (divisible)
- $3678192 \div 2 = 1839096$ (divisible)
- $3678192 \div 3 = 1226064$ (divisible)
- $3678192 \div 6 = 613032$ (divisible)
- $3678192 \div 7 = 525456$ (divisible)
- $3678192 \div 8 = 459774$ (divisible)
- $3678192 \div 9 = 408688$ (divisible)

Since $3678192$ is divisible by each of its digits, the construction is valid.

The final answer is $\boxed{1, 2, 3, 6, 7, 8, 9}$.