Quadratic reciprocity applied

Reciprocity. $\left(\frac{5}{13}\right)=\left(\frac{13}{5}\right)=\left(\frac{3}{5}\right)$ (since $13\equiv3\pmod5$).

Compute $\left(\frac{3}{5}\right)$. Squares mod $5$: $\{1,4\}$. $3$ not a square, so $\left(\frac{3}{5}\right)=-1$.

Conclusion. $\left(\frac{5}{13}\right)=-1$: $5$ is not a quadratic residue mod $13$.

Check. Squares mod $13$: $\{1,4,9,3,12,10\}$; $5$ absent. ✓