13.211. The digits of a certain three-digit number form a geometric progression. If in this number the digits of the hundreds and units are swapped, the new three-digit number will be 594 less than the desired one. If, however, in the desired number the digit of the hundreds is erased and the digits of the resulting two-digit number are rearranged, the new two-digit number will be 18 less than the number expressed by the last two digits of the desired number. Find this number.
## Solution.

Let the desired number have the form $100 x+10 y+z$, where $x, y, z$ form a geometric progression, which means $x z=y^{2}$. From the condition, we get:

$$
\begin{aligned}
& 100 z+10 y+x=100 x+10 y+z-594 \Rightarrow \\
& \Rightarrow x-z=6 ; 10 y+z=10 z+y+18 \Rightarrow y-z=2
\end{aligned}
$$

We have the system of equations

$$
\left\{\begin{array}{l}
x z=y^{2} \\
x-z=6, \\
y-z=2
\end{array}\right.
$$

Answer: 842.