Problem 4. Place digits in the positions of $a$ and $b$, in the four-digit numbers, so that the sum $\overline{323 a}+\overline{b 410}$ is divisible by 9. Determine all possible solutions.
Solution. The sum of the digits in the given expression is $13+a+b$. For the number to be divisible by 9, the sum of its digits must also be divisible by 9, and since $a+b \leq 18$, we have two possibilities, namely $a+b=5$ or $a+b=14$, where $b \neq 0$. Then each of the ordered pairs:

$$
(0,5),(1,4),(4,1),(2,3),(3,2),(5,9),(9,5),(6,8)(8,6) \text { and }(7,7)
$$

is a solution.