Making two digit (or more) mental division easier by using negative remainders

In a chat with Tiger, the following came up:

Work with both positive and negative remainders.

Why would you do this? Well, it keeps the digits you work with down to 0-5, which is easier than to work with 0-9.
What I mean is, that for example if you need to divide by 42, you have probably learned that if the next digit is 9 you would first need to work out 42X9 and then subtract. While this is a consistent system and doable on paper, this is difficult to do in your head.

In my way of calculating the previous digit would be 1 more- a little bit too much - and the remainder becomes negative, so the next digit, instead of 9, becomes -1 and instead of adding numbers I subtract. An example will make this clear.

Let’s do 100/42:

First digit: 2. Subtract in your mind using 2 steps: 2X40 and 2X2:
100 - 80 - 4 = 16. In my mind, I see: 100 -> 20 ->16

  1. Use ‘4’ and realise we end up with a negative remainder. Again 2 steps 4X40 and 4X2 = 160 and 8
    160 - 160 - 8 = -8

Answer so far: 24 R-8
Realise because of the negative remainder that 24 is too big, so it needs to come down a bit. Therefore, in the next step we will be subtracting instead of adding.

Next digit. 24 R-8 -> 240 R-80
To get to the positive again I need to add 2X42 = 84 to the remainder of -80.
Since we now work with a negative remainder I subtract 2 from 240 to get 238. Remainder is 4 and we are back to positive.

Answer so far: 238 R 4 -> 2380 R 40

40 is almost 42. So use 1 and realise we again end up negative. No worries. 40 - 42 = -2.

Answer so far: 2381 R -2 -> 23810 R -20

-20 is not enough to have any factor of 42, so next digit is 0:

Answer so far: 23810 R -20 -> 238100 R -200

We need 5X42 to get into the positive again:
-200 + 200 + 10 = 10
Subtract 5 from 238100 => 238095

Answer so far: 238095 R 10 -> 2380950 R 100

Now since the remainder is 100, which we started with, we are back where we started
Repeat from the top and the answer becomes:

2.38095 238095 238095 238095 238095 238095 238095 238095 238095…

Thank you Kinma, that is a very creative and clever way of doing things. Takes a little practice but it is effective, I appreciate tHat you have taken the time to explain and share this.