jmc

Because $$1+2+\cdots +n=\frac{n(n+1)}{2},$$$1+2+\cdots +n$ divides the positive integer $6n$ if and only if $$\frac{6n}{n(n+1)/2}=\frac{12}{n+1}\text{is an integer.}$$There are $\boxed{5}$ such positive values of $n$, namely, 1, 2, 3, 5, and 11.