To find two positive numbers whose product is 100 and whose sum is a minimum, follow these steps:

1. Let the numbers be $x$ and $y$.

   • We know $x \times y = 100$.

2. Express the sum in terms of one variable.

   • $y = \frac{100}{x}$.
   • Sum, $S = x + y = x + \frac{100}{x}$.

3. Minimize the sum $S$.

   • Differentiate $S$ with respect to $x$: $\frac{dS}{dx} = 1 - \frac{100}{x^2}$.
   • Set the derivative to zero: $1 - \frac{100}{x^2} = 0$.
   • Solve for $x$: $x^2 = 100 \Rightarrow x = 10$.

4. Find $y$ using $x = 10$.

   • $y = \frac{100}{x} = \frac{100}{10} = 10$.

Thus, the two numbers are 10 and 10.