Robert’s Remainder


Robert’s Remainder is defined as the number you build by successive remainders in a classic long division process. What is the quotient, remainder, and Robert’s Remainder of 1234 divided by 5?

Does Robert’s Remainder have any interesting properties?



1234 divided by 5 gives you 246 with a remainder of 4 and a Robert’s Remainder of 234. Here is the work:

Thoughts and Opinions

The namesake of Robert’s Remainder has since graduated from MIT. This page is a reflection on his invention. It is my hope to spread awareness of Robert’s Remainder to as many as possible.

Some say a gifted mind sees new things that are not there. Some say it is the symptom of a mind that is not grounded in reality. The truth is Robert has the bravery to make his imagination real through his art. To him and all brave individuals I am eternally thankful. Sometimes you can be inspiring without even trying.

Robert’s Remainder is very real. It represents a third number recoverable from the long division process. We are familiar with the first from the youngest age we learn about dividing numbers: the quotient. The word “quotient” was invented by some other brave individual hundreds of years ago to simply mean the answer you get when you divide two numbers. I commonly just call this the answer, but in reality it is just an answer.

When you divide numbers, they can “go in evenly” or there can be “something left over.” So it is said, dividing numbers can leave a “remainder.” This word I like a lot more, it says exactly what it means. The remainder when you divide 1234 by 5 is 4. That means, you could divide 1230 by 5 evenly but the 4 remaining cannot divide into 5 parts because 4 is too small. We learned next to write that like a fraction tacked on to the quotient as it is written in the Facts above. Thus, 4 is an answer to “1234 divided by 5.” It is the remainder answer.

When we learn the procedures for dividing, these are reflections on the mental labor of individuals who came before us. There was a time before the recipe for long division. It needed to be codified to help us know more about the numbers which have always existed. We are born ill equipped for numbers, so we need these tools.

234 is an answer to 1234 divided by 5. It has a name to honor the individual brave enough to codify it.

I will leave it as an exercise to the reader to find the interesting properties of this new answer. Here are some questions on this topic:

  • Will the last digits of Robert’s Remainder always be the same as the conventional remainder?
  • When a remainder in a given long division step is recovered that is two or three or more digits, do those get “tacked on,” or do they sum as conventional addition to Robert’s Remainder?
  • What about “Robert’s Reduced Remainder,” formed by successively replacing the dividend with the previous Robert’s Remainder answer? For reference, Robert’s Reduced Remainder for “1234 divided by 5” is 4 by the chain 234 -> 24 -> 4. You may write RRR(1234,5) = RR(RR(RR(1234,5),5),5) = 4 if that suits you.
  • When is Robert’s Reduced Remainder the same as the conventional remainder, and why?
  • Is there a pair of numbers for which Robert’s Reduced Remainder is infinite? If so, what properties do numbers have that lead them to cases where Robert’s Reduced Remainder “blows up” ?
  • Is there a pair of numbers for which Robert’s Reduced Remainder is the same as the original dividend? In other words, is there a pair of numbers for which Robert’s Reduced Remainder “loops” back to the beginning? What properties do these numbers have?

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