$n^2+15n+36=k^2$ (Với $k∈N^*$)
$↔ 4n^2+60n+144=4k^2$
$↔ (2n)^2+2.2n.15+225-81=4k^2$
$↔ (2n+15)^2-4k^2=81$
$↔ (2n+15+2k)(2n+15-2k)=81$
Thử với $\left\{ \begin{array}{l}2n+2k+15=81\\2n-2k+15=1\end{array} \right.$
$↔ \left\{ \begin{array}{l}2n+2k=66\\2n-2k=-14\end{array} \right.$
$↔ \left\{ \begin{array}{l}4n=52\\n-k=-7\end{array} \right.$
$↔ \left\{ \begin{array}{l}n=13\\k=n+7\end{array} \right.$
$↔ \left\{ \begin{array}{l}n=13\\k=20\end{array} \right.$ (thỏa mãn)
Vậy $n=13$
