a) If $d$ is a common divisor of the numbers $12n + 13$ and $13n + 14$, then $d$ divides the numbers $13(12n + 13)$ and $12(13n + 14)$. Then $d$ divides $13(12n + 13) - 12(13n + 14)$, that is $d \mid 1$. So $d = 1$, hence the numbers $12n + 13$ and $13n + 14$ are coprime.

b) The relation $\frac{a}{b} = \frac{12n+13}{13n+14}$ leads to $a(13n+14) = b(12n+13)$, hence $13n+14 \mid b(12n+13)$. Since $12n + 13$ and $13n + 14$ are coprime, $13n + 14$ divides $b$. Since

$b \neq 0$ (it is the denominator of a fraction), there exists $k \in \mathbb{N}^*$ so that $b = k(13n+14)$ and therefore $a = k(12n+13)$.

Replace $a = k(12n + 13)$ and $b = k(13n + 14)$ in $17a + 19b < 2024$ to get
$k(451n + 487) < 2024$. Then $451n + 487 < 2024$, whence $n \le 3$.

For $n = 0$ we get $k \le 4$, so $(a, b) \in \{(13, 14), (26, 28), (39, 42), (52, 56)\}$.
For $n = 1$ we get $k \le 2$, so $(a, b) \in \{(25, 27), (50, 54)\}$.
For $n \in \{2, 3\}$ we get $k = 1$, so $(a, b) \in \{(37, 40), (49, 53)\}$.

In total, there are 8 pairs.