Find the integers $n$ such that the fraction $\frac{3 n+10}{5 n+16}$ is in irreducible form.
We know that the fraction $\frac{3 n+10}{5 n+16}$ is irreducible if and only if $\operatorname{GCD}(3 n+10,5 n+16)=1$. The Euclidean algorithm tells us that

$$
\begin{aligned}
\operatorname{GCD}(3 n+10,5 n+16) & =\operatorname{GCD}(3 n+10,2 n+6) \\
& =\operatorname{GCD}(n+4,2 n+6) \\
& =\operatorname{GCD}(n+4,2) \\
& =\operatorname{GCD}(n, 2)
\end{aligned}
$$

Therefore, we conclude that the fraction $\frac{3 n+10}{5 n+16}$ is irreducible if and only if
$\operatorname{GCD}(n, 2)=1$, that is, if and only if $n$ is odd.