Why does the method for division work?

Question

Why does this method for dividing two integers work? For example:

 127 : 4 = 31,75
−12
  07
  −4
   30
  -28
    20
   −20
     0

Answer 1

You are dividing 127 by 4 using the long division.

$$127/4=$$

$$120/4+7/4=$$

$$30 + 1+3/4=$$

$$31 +0.1(30/4)=$$

$$31+0.1(7+2/4)=$$

$$31.7+0.01(20/4)=31.75$$

Answer 2

One simple method to divide an integer $b$ by $a$ is to subtract $a$ repeatedly from $b$ and count how many times you do this. However this becomes tedious one at a time so we use a shortcut to counting a whole bunch of them at once - subtracting off some multiple of $a$ that are smaller than $b$. If you do this in the most efficient way using your memory of the times tables you come up with the standard long division algorithm. You can vary how you do the counting though and it works perfectly well as long as you keep an accurate count of how many $a$'s you subtract off at each stage.