Let's go through each of the two questions and solve them one by one.

Question 1

Find two positive integers whose difference is 50 and whose product is minimum.

Let the two integers be $x$ and $y$ such that:

1. $x - y = 50$
2. We want to minimize the product $P = x \cdot y$.

From the first condition, we can express $x$ in terms of $y$ as: $x = y + 50$

Substituting this into the product $P$: $P = (y + 50) \cdot y = y^2 + 50y$

To find the minimum value, we take the derivative of $P$ with respect to $y$ and set it to zero: $\frac{dP}{dy} = 2y + 50 = 0$ $2y = -50$ $y = -25$

Since $y$ must be positive (as we’re looking for positive integers), we cannot use $y = -25$. Instead, let's try possible integer values close to the calculated result or solve through other integer-based constraints.

Therefore, after evaluating positive integers closely around the theoretical result