What is the value of $xyz$?

Question

Given an expression as :

$$xyz+xyz+xyz=zzz$$ where $x,y,z$ are integers and $xyz$ represents a number for example $236$ (not to be confused with $x\times y\times z$), what is the number $xyz$?

Answer 1

From $3z\equiv z\pmod{10}$ we find $2z\equiv 0\pmod {10}$, so $z=0$ (which is absurd per right hand side) or $z=5$. So we must have $xyz=\frac{555}{3}$.

Answer 2

From the last digit we have $z+z+z=z$ or $z+z+z=10+z$ or $z+z+z=20+z$, so $z=0$ or $z=5$.

If $z=0$ then we have $y+y+y=0$ or $10+0$ for the second digit, so $z=0,y=0$ and similarly $x=0$.

If $z=5$ then there's a carry to the 2nd digit and $y+y+y+1$ = $5$ (no solution) or $y+y+y+1=14$ (no solution) or $y+y+y+1=25$, i.e. $y=8$, with a carry of $2$ to the first digit. Then $x+x+x+2=5$, so $x=1$.

So $xyz$ = $000$ or $185$.