Let $n$ be a positive integer such that one of the roots of the quadratic equation $4x^2-\left(4\sqrt{3}+4\right)x+\sqrt{3}n-24=0$ is an integer. Find the value of $n$.

If $\alpha , \beta$ are roots of the equation $\lambda \left(x^2-x\right)+x+5=0.$ If ${\lambda }_1$ and ${\lambda }_2$ are the two values of $\lambda$ for which the roots $\alpha ,\beta$ are connected by the relation $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=4,$ then the value of $\left(\frac{\frac{{\lambda }_1}{{\lambda }_2}+\frac{{\lambda }_2}{{\lambda }_1}}{14}\right)$ is