why do we divide 60% into our AFM landing distance number?
Because that is how you work the problem backwards, to make it easier. But you don't have to do it that way... follow me through
Forget about any specific airport or runway length. Lets talk temp & field elevation, and say your unfactored landing distance is 2500'. Since you must stop within 60% of some runway, what is the minimum length you can accept? 4167', we've answered that.
Lets work it the other way... you want to land at an airport with a 7000' runway. You can either go to the book and look at the factored distance and see if it's equal to or less than 7000'. OR you could find 60% of 7000', which is 4200' and see if the unfactored distance required is less.