When does division with remainder exist?

Added: 2026-05-05

[00:00:00]Quick question. What would a third grader say if you asked them to divide 17 by three? Well, likely they would say that you get five with a remainder of two. Because that's the kind of division that you learn in the beginning of elementary school, division with remainder. I bet you could twist your brain around and convince yourselves that if you divide these two complex integers, you would get a quotient of 1 - 2i and a remainder of 2 + i. But what about in other extensions of the integers? Like for instance, could you divide 1 + the square root of -5 by 2 and get a meaningful quotient and remainder? Well, I just recorded a whole video about this that's going on the channel very soon. So, make sure and subscribe so you don't miss it.