The number of positive integers k such that the constant term in the binomial expansion of (2 x^3+3/x^k)^12, x ≠ 0 is 2^8 · ℓ, where ℓ is an odd…

The number of positive integers $\displaystyle k$ such that the constant term in the binomial expansion of $\displaystyle \left(2 x^3+\frac{3}{x^k}\right)^{12}, x \neq 0$ is $\displaystyle 2^8 \cdot \ell$, where $\displaystyle \ell$ is an odd integer, is \_\_\_\_.