## Task B-3.5.

What values can the digits $a$ and $b$ take so that the sum of the numbers $\overline{29 a 8}$ and $\overline{342 b}$ is divisible by 18?
## Solution.

The sum of the given numbers is written as

$$
S=2908+10 a+3420+b=6318+10+10 a+b=351 \cdot 18+10 a+b+10 . \quad 2 \text { points }
$$

Then $S$ is divisible by 18 if the number $N=10 a+b+10$ is divisible by 18.

Since $a, b \in\{0,1,2, \ldots, 9\}$, the smallest value that the number $N$ can have is 10, and the largest is 109.

Moreover, it is a multiple of the number 18, so $N$ must be within the set $\{18,36,54,72,90,108\}$. 1 point Then $10 a+b \in\{8,26,44,62,80,98\}$, that is,

$$
(a, b) \in\{(0,8),(2,6),(4,4),(6,2),(8,0),(9,8)\}
$$