Which positive three-digit numbers satisfy the following two equations:

$$
\begin{aligned}
& 17 x+15 y-28 z=61 \\
& 19 x-25 y+12 z=31
\end{aligned}
$$
(1) Adding five times (1) and three times (2), and dividing both sides by 2, we get

$$
71 x-52 z=199
$$

from which

$$
\begin{aligned}
& z=\frac{71 x-199}{52}=x-3+\frac{19 x-43}{52}=x-3+t \\
& x=\frac{52 t+43}{19}=2 t+2+\frac{14 t+5}{19}=2 t+2+u \\
& t=\frac{19 u-5}{14}=u+\frac{5 u-5}{14}=u+5 \cdot \frac{u-1}{14}=u+5 v \\
& u=14 v+1
\end{aligned}
$$

Substituting back

$$
\begin{aligned}
t & =14 v+1+5 v=19 v+1 \\
x & =38 v+2+2+14 v+1=52 v+5 \\
z & =52 v+5-3+19 v+1=71 v+3
\end{aligned}
$$

From (1)

$$
\begin{gathered}
y=\frac{61-17 x+28 z}{15}=\frac{61-17(52 v+5)+28(71 v+3)}{15}=\frac{1104 v+60}{15}= \\
=\frac{368 v+20}{5}=73 v+4+\frac{3 v}{5}=73 v+4+3 w \\
v=5 w
\end{gathered}
$$

Substituting back again

$$
\begin{aligned}
& y=368 w+4 \\
& x=260 w+5 \\
& z=355 w+3
\end{aligned}
$$

For $x, y$, and $z$ to be three-digit numbers, it is necessary and sufficient that $w=1$ or $w=2$.
For $w=1$:
$x=265$,
$y=372$
$z=358$
For $w=2$:
$x=525$,
$y=740$,
$z=713$.

Tamás Fuchs (Bp., II., Rákóczi g. III. o. t.)