320. Divisor. For what values of $N$ does the number $N^{2}-71$ divide by $7 N+55$?
320. If $\frac{N^{2}-71}{7 N+55}=M$ is an integer, then

$$
N^{2}-7 M N-(55 M+71)=0
$$

When we solve this quadratic equation for $N$, we will find that the expression under the radical must be a perfect square. Since the inequalities hold

$$
\begin{aligned}
& (7 M+15)^{2}=49 M^{2}+210 M+225<49 M^{2}+220 M+284< \\
& \quad<49 M^{2}+238 M+289=(7 M+17)^{2}
\end{aligned}
$$

the expression under the radical must equal $(7 M+16)^{2}$. From this, $M=7$, and $N=57$ or $(-8)$.

[D. Silverman, M. M., 44, 56 (January 1971).]