7. [6] A triple of integers $(a, b, c)$ satisfies $a+b c=2017$ and $b+c a=8$. Find all possible values of $c$.
Answer: $\quad-6,0,2,8$
Add and subtract the two equations to find
$$
\begin{array}{l}
(b+a)(c+1)=8+2017, \\
(b-a)(c-1)=2017-8 .
\end{array}
$$

We see that $c$ is even and then that every integer $c$ with $c+1|2025, c-1| 2009$ works. We factor and solve.

The full solutions are $(2017,8,0),(-667,1342,2),(-59,-346,-6),(-31,256,8)$.