You are correct that you are looking for two numbers, $x$ and $y$ that satisfy $x-y=100$, or $y=x-100$.
But you seem to have forgotten about the second part of the problem, which says:
... whose product is a minimum.
That is, among all numbers $x$ and $y$ that satisfy $x-y=100$, you are looking for two with $xy$ smallest possible.
For instance, one possibility would be $x=100$, $y=0$; then $xy=0$. Or you could take $x=90$, $y=-10$, in which case $xy=-90$; clearly, $x=90$, $y=-10$ is a "better" choice than $x=100$, $y=0$, because you are looking for a choice that makes the product as small as possible.
So you are trying to minimize (optimization problem alert!) the product $xy$. Since $y=100-x$, that means that you are looking for the minimum of the function $p(x) = xy = x(100-x).$
So you want to find the absolute minimum of $p(x)$. Can you take it form here?