Task 3. Add a digit to the left and a digit to the right of the number 10 so that the resulting four-digit number is divisible by 9 and by 4.
Solution. For the new number $\overline{a 10 b}$ to be divisible by 4, it needs to end in 00, 04, or 08, i.e., $b \in\{0,4,8\}$. For the number to be divisible by 9, the sum of its digits must be divisible by 9, i.e., $a+b+1$ must be divisible by 9. For $b=0$, the sum $a+b+1=a+1$ can only be 9. Then, $a=8$ and the number is 8100. For $b=4$, the sum $a+b+1=a+5$ can only be 9. Then, $a=4$ and the number is 4104. Finally, for $b=8$, the sum $a+b+1=a+9$ can only be 18. Then, $a=9$ and the number is 9108.