20. Find all nonnegative integer solutions of the system
$$
\begin{array}{l}
5 x+7 y+5 z=37 \\
6 x-y-10 z=3 .
\end{array}
$$
(ans. $(x, y, z)=(4,1,2)$

Eliminating $z$ by multiplying the first equation by 2 and taking the sums, we obtain $16 x+13 y=77$. This is equivalent to $16(x-4)+13(y-$ $1)=0$, hence $(x, y, z)=(4,1,2)$ is a solution. All other solutions are given by $x=4+16 t, y=1-13 t, t$ an integer. $x \geq 0 \Rightarrow t \geq 0$. This implies $y \leq 0 \Rightarrow t \leq 0$, hence $t=0$.)