For which values of $n \in \mathbb{N}$ is the fraction $\frac{2 n^{2}+11 n-18}{n+7}$ irreducible?
We want to calculate the GCD of $2 n^{2}+11 n-18$ and $n+7$, so let's try the Euclidean algorithm:

$$
2 n^{2}+11 n-18=(n+7) \times(2 n-3)+3
$$

Therefore, the GCD we are looking for is a divisor of 3. But 3 divides $2 n+7$ if and only if $n \equiv 2[3]$, so if $n \equiv 0$ or $1[3]$, then the fraction is irreducible. Conversely, we verify that if $n \equiv 2[3]$ then the fraction can be simplified by 3.