The number of positive integers k such that the constant term in...

For ${\left(2{x}^{3} + \dfrac{3}{{x}^{k}}\right)}^{12}$

${T}_{r + 1} = Cr 12 {\left(2{x}^{3}\right)}^{12 - r} \cdot {\left(\dfrac{3}{{x}^{k}}\right)}^{r} = Cr 12 \cdot {2}^{12 - r} \cdot {3}^{r} \cdot {x}^{36 - 3r - kr}$

For the term to be independent of $x$,

$36 - 3r - kr = 0$ $⇒ r = \dfrac{36}{k + 3}$

Here $k$ can be $0, 1, 3, 6, 9$ for $r$ to be a whole number.

But for $Cr 12 \cdot {2}^{12 - r} \cdot {3}^{r}$ to be in the form of ${2}^{8} \cdot l$

Only $k = 3, 6$ satisfies the above relation.