Roots' relation

Because $\alpha$ and $\beta$ are the roots of the equation $2x^2-mx+8=0$ , we have: $\left\{\begin{array}{l}\alpha+\beta=\dfrac{m}{2}\\\alpha\beta=4\end{array}\right.$ Otherwise, the roots of the equation $5x^2-10x+5n=0$ are $\dfrac{1}{\alpha}$ and $\dfrac{1}{\beta}$ , we get: $\left\{\begin{array}{l}\dfrac{1}{\alpha}+\dfrac{1}{\beta}=2\\\dfrac{1}{\alpha}.\dfrac{1}{\beta}=n\end{array}\right.$ Thus, $mn=\dfrac{2(\alpha+\beta)}{\alpha\beta}=2\left(\dfrac{1}{\alpha}+\dfrac{1}{\beta}\right)=\boxed{4}$ .

Bonus : We can find that $n=\dfrac{1}{4}; m=16$ .