## Task B-4.3.

Determine all three-digit numbers that in base 9 have the representation $\overline{x y z}_{9}$, and in base 11 have the representation $\overline{z y x}_{11}$.
## Solution.

Digits $x, y, z$ can take values from the set $\{0,1,2,3,4,5,6,7,8\}$ and $x, z \neq 0$.

By expanding in base, we have:

$$
\begin{aligned}
81 x + 9 y + z & = 121 z + 11 y + x \\
2 y & = 80 x - 120 z \\
y & = 40 x - 60 z = 20(2 x - 3 z) .
\end{aligned}
$$

The only possibility (since they are digits) is $y=0$, and $2 x = 3 z$, which leads to

$$
\begin{array}{lll}
1^{\circ} & x=3, z=2 \Longrightarrow 302_{9}=203_{11} & 1 \text{ point} \\
2^{\circ} & x=6, z=4 \Longrightarrow 604_{9}=406_{11} . & 1 \text{ point}
\end{array}
$$