## Task 2 - 230912

Determine all two-digit natural numbers $x$ and $y$ that satisfy the following conditions:

(1) The number $y$ is obtained from $x$ by swapping the two digits.

(2) It holds that $x+y=121$.
Condition (1) is satisfied if and only if

$$
x=10 a+b, \quad y=10 b+a
$$

with two natural numbers $a, b$ for which $1 \leq a \leq 9$ and $1 \leq b \leq 9 \quad$ (4) holds. This makes (2) satisfied if and only if $a$ and $b$ satisfy, in addition to (4),

$$
10 a+b+10 b+a=121 \quad \text { or, equivalently, } \quad 11 \cdot(a+b)=121 \quad, \quad a+b=11
$$

Since (4) and (5) are satisfied exactly by the natural numbers $a, b$ given in the following table, it follows from (3) that exactly the subsequently specified numbers $x, y$ have the required properties.

| $a$ | $b$ | $x$ | $y$ | $a$ | $b$ | $x$ | $y$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2 | 9 | 29 | 92 | 3 | 8 | 38 | 83 |
| 4 | 7 | 47 | 74 | 5 | 6 | 56 | 65 |
| 6 | 5 | 65 | 56 | 7 | 4 | 74 | 47 |
| 8 | 3 | 83 | 38 | 9 | 2 | 92 | 29 |