Task 4. Before and after the number 10, write one digit each so that the resulting number is divisible by 36 and the quotient of that number divided by 36 is a number written with different digits from the given number.
Solution. Let's denote the desired number as $\overline{a 10 b}$. This number must be divisible by 36, which means it must be divisible by 4 and by 9. From the condition that the number is divisible by 4, it follows that the last digit can be 0, 4, or 8. From the condition that the number is divisible by 9, it follows that $9 \mid(a+1+b)$, which means $a \in\{4,8,9\}$. Therefore, the possible numbers are 4104, 8100, and 9108. Since $4104: 36=114$, $8100: 36=225$, and $9108: 36=253$, it follows that the desired number is 9108.