We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard group and it's Class group.
Should I expect this result to make sense before proving it? Is there a geometric significance to UFD's? I do not know of any other significant theorem that needs as its assumptions factoriality. Why is it suddenly important here and what does it have to do with divisors/line bundles?