Gábor wants to call Péter on the phone, but he only remembers that the three-digit phone number is divisible by 13, and the middle digit is equal to the arithmetic mean of the outer digits. Which numbers should Gábor call to speak with Péter?
Let the digits of the phone number $N$ be $x, y, z$. According to Gábor's memory,

\[
\begin{aligned}
100 x + 10 y + z & = 13 n \quad \text{and} \\
x + z & = 2 y
\end{aligned}
\]

where $n$ is an integer. Substituting $z$ from (2) into (1), we get $99 x + 12 y = 13 n$. The left side can be written as:

\[
99 x + 12 y = 104 x + 13 y - 5 x - y = 13(8 x + y) - 5 x - y
\]

Substituting this and solving for $5 x + y$,

\[
5 x + y = 13(8 x + y - n) = 13 t
\]

where $t$ is an integer. The left side of (3) cannot be greater than if $x$ and $y$ are both 9, so $13 t \leq 5 \cdot 9 + 9 = 54$, thus only $t = 0, 1, 2, 3, 4$ are possible. Solving for $y$ from (3) and $z$ from (2): $y = 13 t - 5 x, z = 26 t - 11 x$. Since $z$ is a digit, for $t = 0, 1, 2, 3, 4$, $x$ can only be $0, 2, 4, 7, 9$ respectively, thus the following possibilities arise:

\[
\begin{aligned}
& t = 0, \quad x = 0, \quad y = 0, \quad z = 0, \quad N_{0} = 000 \\
& t = 1, \quad x = 2, \quad y = 3, \quad z = 4, \quad N_{1} = 234 \quad (= 13 \cdot 18) \\
& t = 2, \quad x = 4, \quad y = 6, \quad z = 8, \quad N_{2} = 468 \quad (= 2 N_{1}) \text{; } \\
& t = 3, \quad x = 7, \quad y = 4, \quad z = 1, \quad N_{3} = 741 \quad (= 13 \cdot 57) \\
& t = 4, \quad x = 9, \quad y = 7, \quad z = 5, \quad N_{4} = 975 \quad (= 13 \cdot 75)
\end{aligned}
\]

The phone number consisting entirely of zeros can be ignored, as it is unlikely that Gábor would have noted its divisibility by 13, even if the telephone exchange had assigned it to a subscriber (it is generally reserved for internal use). Therefore, Gábor needs to try at most four phone numbers.