Confusion regarding voltage division
My answer is how they can just ignore R3 when making the voltage
  division equations in the first place.


In order to make a judgement on this circuit, various techniques can be used and, it is advantageous to the person solving the problem, to choose the method that delivers an answer in the shortest time. The most sensible first method to try (as a thought experiment) is any method that attempts to compare the voltage at node VB with that at node VC with R3 disconnected.

Clearly, within a couple of seconds, an experienced engineer will see that those node voltages are equal and, the impact of this is that no current can flow even if R3 were a short circuit.

You don't even need to form the equations; it's enough to see that the ratio of R4 to R5 is the same as R2 to R6.