1. We are given two quadratic equations: $$2x^2 - mx + 8 = 0$$ with roots $$\alpha$$ and $$\beta$$. $$5x^2 - 10x + 5n = 0$$ with roots $$\frac{1}{\alpha}$$ and $$\frac{1}{\beta}$$. 2. By the properties of quadratic equations, the sum and product of roots for each equation are: For the first equation: $$\alpha + \beta = \frac{m}{2}$$ and $$\alpha\beta = \frac{8}{2} = 4$$. For the second equation: $$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{-(-10)}{5} = 2$$ and $$\frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{5n}{5} = n$$. 3. Express $$\frac{1}{\alpha} + \frac{1}{\beta}$$ in terms of $$\alpha$$ and $$\beta$$: $$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{\frac{m}{2}}{4} = \frac{m}{8}$$. 4. Set the sum from the second equation equal to this: $$2 = \frac{m}{8} \implies m = 16$$. 5. For the product of the roots of the second equation: $$n = \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha\beta} = \frac{1}{4}$$. 6. Finally, compute $$mn$$: $$mn = 16 \times \frac{1}{4} = 4$$. Therefore, $$mn = 4$$.