## Task B-1.3.

Determine all natural numbers $n$ for which the fraction $\frac{35}{2 n-5}$ is an integer.
## Solution.

A fraction will be an integer if and only if the numerator is divisible by the denominator.

This means that the denominator $2 n-5$ can take the values:

$2 n-5 \in\{-35,-7,-5,-1,1,5,7,35\}$.

Then $2 n \in\{-30,-2,0,4,6,10,12,40\}$, 1 point

and $n \in\{-15,-1,0,2,3,5,6,20\}$.

1 point

Since $n$ is a natural number, $n \in\{2,3,5,6,20\}$.

## Note:

If the student lists all the required numbers without any explanation, they can receive a maximum of 2 points. If they include 0 and/or negative numbers, they get 1 point, and for everything else without a process, 0 points. If the student has all the solutions and has shown that the desired fraction is an integer for each of them by substitution, assign 3 points. To receive all 6 points, the student must show that the obtained solutions are the only possible solutions, that is, list all divisors of the number 35 (as described by the previous reasoning). If they only considered positive divisors and thus obtained the solution 3, 5, 6, 20, assign 4 points.