## Task 3 - 180823

Determine all two-digit natural numbers with the following property:

If you add $2$ to the desired number, you get three times the number that results from swapping the digits of the original number.
Assume there is such a number. Then it has the form $10 x+y$, where $x$ and $y$ are natural numbers with $x, y \leq 9$. For these, the following holds:

$$
10 x+y+2=3(10 y+x) \quad \text { thus } \quad y=\frac{7 x+2}{29}
$$

Since $y$ is a natural number, $7 x+2$ must be a multiple of 29. Given $0<x \leq 9$, it follows that $2<7 x+2 \leq 65$; therefore, only the cases $7 x+2=29$ and $7 x+2=58$ are possible.

$7 x+2=29$ is not solvable in natural numbers. From $7 x+2=58$ it follows that $x=8$; for $y$ one then gets 2.

Thus, the number 82 can at most satisfy the conditions of the problem. It actually does; because $82+2=84=3 \cdot 28$.