Find the largest natural number, all digits in the decimal representation of which are different and which

is reduced by 5 times if the first digit is erased.
Given $\overline{a A}=5 A$ (where $A$ is the number composed of all digits except the first, and $a$ is the first digit). Let $n$ be the number of digits in $\overline{a A}$. Then

$4 A=a \cdot 10^{n-1}$, which means $A=25 a \cdot 10^{n-3}$.

If $n>4$, then the number $A$, and thus the desired number, has two identical digits (two zeros at the end). If $n=4$, then $A=250 a$. Clearly, the larger $a$ is, the larger the original number. For $a \geq 4$, the number $250 a$ consists of four digits, not three. When $a=3$, we get $A=750$.

## Answer

3750.