[tex]\displaystyle\\
x,~y,~z~~\text{ sunt cifre}\\\\
n=\overline{x,\!(y)}+\overline{y,\!(z)}+\overline{z,\!(x)}\\\\
n= \frac{\overline{xy}-x }{9}+ \frac{\overline{yz}-y }{9}+ \frac{\overline{zx}-z }{9}\\\\
n= \frac{10x+y-x }{9}+ \frac{10y+z-y }{9}+ \frac{10z+x-z }{9}\\\\
n= \frac{9x+y}{9}+ \frac{9y+z}{9}+\frac{9z+x}{9}\\\\
n= \frac{9x}{9}+ \frac{y}{9} + \frac{9y}{9}+ \frac{z}{9}+\frac{9z}{9}+ \frac{x}{9}\\\\
n= \frac{9x}{9}+ \frac{9y}{9}+\frac{9z}{9}+ \frac{y}{9} + \frac{z}{9}+ \frac{x}{9}
[/tex]
[tex]\displaystyle\\
n= \frac{9x+9y+9z}{9}+ \frac{x+y+z}{9} \\\\
n= \frac{9(x+y+z)}{9}+ \frac{x+y+z}{9} \\\\
n=(x+y+z)+\frac{x+y+z}{9}\\\\
n \in N
(x+y+z) \in N
\Longrightarrow~~\frac{x+y+z}{9}\in N~~\Longrightarrow~~(x+y+z)~\vdots~9 \\\\
\text{(Problema nu spune ca x; y; z sunt cifre diferite.)}\\\\
\Longrightarrow~~(x+y+z) = 9\cdot k\\ M_9 = \{0;~9;~18;~27\} \\\\
\Longrightarrow~~k \in =\{0;~1;~2;~3\}[/tex]
[tex]\displaystyle\\
\text{Rezulta solutiile:}\\\\
n_1 = (x+y+z)+\frac{x+y+z}{9} = 0 + \frac{0}{9} =0+0=\boxed{0}\\
n_1 ~~\text{este solutie daca acceptam scrierea numarului zecimal:}~0,\!(0)=0 \\\\
n_2 = (x+y+z)+\frac{x+y+z}{9} = 9 + \frac{9}{9} =9+1=\boxed{10}\\
\text{Pentru }n_2 \text{ sunt multe combinatii de cifre a caror suma = 9.}\\\\
n_3 = (x+y+z)+\frac{x+y+z}{9} = 18 + \frac{18}{9} =18+2=\boxed{20}\\
\text{Pentru }n_3 \text{ sunt multe combinatii de cifre a caror suma = 18.}\\\\
[/tex]
[tex]\displaystyle\\
n_4 = (x+y+z)+\frac{x+y+z}{9} = 27 + \frac{27}{9} =27+3=\boxed{30}\\
n_4 ~~\text{este solutie daca acceptam scrierea numarului zecimal:}~9,\!(9)=10\\\\
\boxed{n \in \{0;~10;~20;~30\}} [/tex]
