Lets try a game:
I'm thinking of a number, lets call it my private factor.
I tell you that it is less than some other number, which we'll call my public key.
For any number you choose, I'll tell you whether your choice is above or below my private factor.
How long will it take you to guess my factor?
Lets try. my public key is 24.
Is the factor above 10? No. Is the factor above 5? Yes. Is the factor above 7? No. Is it 6? Yes.
And look: 24 / 6 = 4 ! You guessed my private key, and you happen to have factored my public key at the same time! Wow!
You only found a single set of factors for your public key (ie 3,8 also work) and if I had asked "is the number 6?" as my first question then I would have had it in 1 single guess which does *NOT* qualify as factoring your key.
You may not think that you are talking about factoring, but factoring is a subset of what you are discussing.
the fact it is a subset of what I am talking about means that there are some issues (and possibly an algorithm or two) that are outside of the purvue of a discussion limited to simply factoring. The horizon has been expanded.