## Task A-1.1.

If $a$ and $b$ are natural numbers, then $\overline{\overline{a . b}}$ is the decimal number obtained by writing a decimal point after the number $a$ and then the number $b$. For example, if $a=20$ and $b=17$, then $\overline{\overline{a . b}}=20.17$ and $\overline{\overline{b . a}}=17.2$.

Determine all pairs $(a, b)$ of natural numbers for which $\overline{\overline{a \cdot b}} \cdot \overline{\overline{b . a}}=13$.
## Solution.

If $(a, b)$ is a solution, then $(b, a)$ is also a solution, and vice versa. Therefore, assume that $a \geqslant b$.

If $a \geqslant 10$, then it is necessary that $b=1$, and the possibilities for $a$ are only 10, 11, and 12.

By direct verification, we see that in these cases the product is not 13.

Therefore, the numbers $a$ and $b$ are both less than 10, i.e., they are digits. The desired condition is equivalent to the condition

$$
\overline{a b} \cdot \overline{b a}=1300
$$

Since the product ends in the digit 0, with the assumption $a \geqslant b$, the only possibility is $a=5, b=2$. By verification, we see that $52 \cdot 25=1300$, i.e., these numbers indeed satisfy the condition of the problem. Therefore, the solutions are $(a, b)=(2,5)$ and $(a, b)=(5,2)$.