1. Determine all integers $n$ for which $\frac{5 n-23}{n-7}$ is an integer.
Solution. The given fraction can be written in the form

$$
\frac{5 n-23}{n-7}=\frac{5 n-35+12}{n-7}=5+\frac{12}{n-7}
$$

Therefore, the given expression is an integer if and only if 12 is divisible by $n-7$.

We have

$$
n-7 \in\{ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\}
$$

or equivalently

$$
n \in\{-5,1,3,4,5,6,8,9,10,11,13,19\} . \quad \text { (6 points) }
$$

## Notes.

- A student who solves it this way but forgets the cases

$n-7 \in\{-1,-2,-3,-4,-6,-12\}$, and thus obtains only 6 solutions, gets a maximum of 15 points.

- A student who finds some (but not all) solutions by trial gets 1 point for each solution found.
- Exceptionally, a student who proves that there are no other solutions (if they test all $n \in\{-5,-4, \ldots, 19,20\}$ and prove that $n>20$ and $n<-5$ lead to a contradiction) gets the maximum number of points.