"Has the same remainder mod 3" partitions the integers into three classes. Why is the quotient construction, treating each class as a single object, one of the most reused moves in mathematics?

C. Because it builds a smaller structure where the distinction has gone

You declare a distinction irrelevant and build a world where it no longer exists, and the operations survive, because addition was well defined on classes. A clock face is ℤ/12ℤ, and the same move constructs the rationals from pairs of integers and quotient groups throughout algebra.