>he should instantly know the pair if both numbers are prime

That should actually be "if the product of their respective smallest prime factors is over 100". 7x11 can't be ruled out since 1x77 also produces 77, whereas 49x17 and Nx53 can be ruled out.

You can also rule out some of the larger squares, e.g. 25x25 and 64x64, so there's probably still a better phrasing for that

Edit: Can also rule out 1xPrime