smc

Let $p$ and $q$ be the roots of $x^2+kx+36.$ By Vieta's Formulas, we have $p+q=-k$ and $pq=36.$ It follows that $p$ and $q$ must be distinct factors of $36.$ The possibilities of $\{p,q\}$ are $$\pm \{1,36\},\pm \{2,18\},\pm \{3,12\},\pm \{4,9\}.$$ Each unordered pair gives a unique value of $k.$ Therefore, there are $\boxed{8}$ values of $k,$ namely $\pm 37,\pm 20,\pm 15,\pm 13.$