Task A-2.1. (4 points)

Determine all natural numbers $n$ for which $\frac{n-1}{n-5}$ is an integer.
## First Solution.

Since

$$
\frac{n-1}{n-5}=\frac{n-5+4}{n-5}=1+\frac{4}{n-5}
$$

the number $\frac{n-1}{n-5}$ is an integer if and only if $\frac{4}{n-5}$ is also an integer,

which holds if and only if 4 is divisible by $(n-5)$,

i.e., if and only if

$$
n-5 \in\{-4,-2,-1,1,2,4\}
$$

Finally, the desired numbers are:

$$
n \in\{1,3,4,6,7,9\}
$$

If the student does not pay attention to the equivalence, they must verify the results.

For grading in such a case, see the second solution.