Once we have established that a relationship exists, it is no longer necessary to memorize a mathematical formula.
The conversion factor works because of that relationship, not because it is equal to one. Rather it is a physical quantity (or the reciprocal of a physical quantity) which is related to the two other quantities we are interconverting. A conversion factor is not unity, however. Using a conversion factor is very similar to using a unity factor-we know the factor is correct when units cancel appropriately.
It is clear that square grams per cubic centimeter are not the units we want.