## Task 2 - 140622

Klaus claims that a natural number $z$ written by him has the following properties:

(1) If two suitable digits of $z$ are swapped, the resulting number $z^{\prime}$ is 198 larger than $z$.

(2) The sum of $z$ and $z^{\prime}$ is 13776.

Determine whether there is exactly one number $z$ with the properties mentioned by Klaus! If this is the case, determine this number!
If there is a number $z$ of the mentioned kind, then for it and the number $z^{\prime}$, the following holds:

(1) $z^{\prime}=198+z$ and

(2) $z+z^{\prime}=13776$

From (1) and (2) it follows: $z+198+z=13776$, from which $2 z=13776-198=13578$, thus $z=6789$. Therefore, only this number can have the mentioned properties.

Indeed, Klaus's statements now apply to this number and $z^{\prime}=6789+198=6987$, since $z^{\prime}$ can be obtained from $z$ by swapping the digits 7 and 9 in $z$.

Therefore, the number $z=6789$ is the only one that has the properties mentioned by Klaus.