58th Ukrainian National Mathematical Olympiad

Clearly, $0$ is not one of the digits. In order to be divisible by $5$ last digit has to be $5$, but the number will not be divisible by $2$, $4$, $6$ and $8$, so it will consist of more digits if $5$ is not one of them. But then if all other digits are present, it is not divisible by $3$, $6$ and $9$. Therefore, one more digit has to be taken from the number in order for it to be divisible by $3$. If so, the following cases are possible:

Case 1. If either $1$ or $7$ are taken, then number is divisible by $3$ and $6$, but not by $9$, thus digit $9$ is not present and the number has $6$ digits;

Case 2. If $4$ is taken, then it is possible to arrange digits that are left so that each digit divides the number, so the number has $7$ digits (all except $0$, $4$ and $5$).

Therefore, the number has $7$ digits: $1$, $2$, $3$, $6$, $7$, $8$, $9$. Our goal is to make it the largest possible.

If the number starts with $987$, then digits $1$, $2$, $3$, $6$ can make numbers divisible by $8$, those are $6312$, $1632$, $2136$ to $3216$. But then any of the numbers is not divisible by $7$. If the number starts with $3216$, then digits $1$, $2$, $3$ form a number $312$ that is divisible by $8$. Moreover, $9867312$ is divisible by $7$, therefore, it is the largest number that satisfies given conditions.