We look at the division of 4177 by 81 
We're looking to find how many times 81 goes into 4177.

Going ahead to step 1:
The number 81 fits into 4 0 times, leaving a remainder of 4.
The number 0 becomes the next digit in our result. 
Result so far: 0.0
Subtracting 0 from 4 leaves us with 4.
Grab the next digit (1) from the dividend, add it to 4, then carry on: 41 / 81

Let's proceed to step 2:
When dividing 41 by 81, we get 0 with a remainder of 41.
Write down 0 as next digit of the result. 
Result so far: 0.0
Deduct 0 from 41 and we're left with 41.
Append the next digit (7) from the dividend to 41 and continue with: 417 / 81

Moving on to step 3:
If we divide 417 by 81, we get 5 and a remainder of 12.
Write down 5 as next digit of the result. 
Result so far: 5.0
Deduct 405 from 417 and we're left with 12.
Fetch the next digit (7) from the dividend, attach it to 12 and continue: 127 / 81

Let's proceed to step 4:
81 goes into 127 1 times with a remainder of 46.
Use 1 as the next digit of our solution. 
Result so far: 51.0
The remainder is 46 after subtracting 81 from 127.
Since there are no more digits in the dividend, we add a zero to the remainder making it 460, and continue the process.

On to step 5:
When dividing 460 by 81, we get 5 with a remainder of 55.
Put 5 as the next digit of the answer. 
Result so far: 51.5
If we subtract 405 from 460, we get 55.
Since there are no more digits in the dividend, we add a zero to the remainder making it 550, and continue the process.

On to step 6:
81 goes into 550 6 times with a remainder of 64.
Use 6 as the next digit of our solution. 
Result so far: 51.56
Deduct 486 from 550 and we're left with 64.
Since there are no more digits in the dividend, we add a zero to the remainder making it 640, and continue the process.

Going ahead to step 7:
640 divided by 81 is 7 with a remainder of 73.
Put 7 as the next digit of the answer. 
Result so far: 51.567
Subtracting 567 from 640 leaves us with 73.
Since there are no more digits in the dividend, we add a zero to the remainder making it 730, and continue the process.

Going ahead to step 8:
81 can be fit into 730 9 times, resulting in a remainder of 1.
Put 9 as the next digit of the answer. 
Result so far: 51.5679
Subtract 729 from 730 to get 1.
Since there are no more digits in the dividend, we add a zero to the remainder making it 10, and continue the process.

On to step 9:
The number 81 fits into 10 0 times, leaving a remainder of 10.
Put 0 as the next digit of the answer. 
Result so far: 51.5679
Subtracting 0 from 10 leaves us with 10.
Since there are no more digits in the dividend, we add a zero to the remainder making it 100, and continue the process.

Advancing to step 10:
If we divide 100 by 81, we get 1 and a remainder of 19.
The next digit of our result is 1. 
Result so far: 51.567901
The remainder is 19 after subtracting 81 from 100.
Since there are no more digits in the dividend, we add a zero to the remainder making it 190, and continue the process.

Moving on to step 11:
If we divide 190 by 81, we get 2 and a remainder of 28.
Record the quotient 2 as the next digit in the result. 
Result so far: 51.5679012
If we subtract 162 from 190, we get 28.
Since there are no more digits in the dividend, we add a zero to the remainder making it 280, and continue the process.

On to step 12:
When dividing 280 by 81, we get 3 with a remainder of 37.
Write down 3 as next digit of the result. 
Result so far: 51.56790123
If we take 243 away from 280, we end up with 37.
Since there are no more digits in the dividend, we add a zero to the remainder making it 370, and continue the process.

The quotient of the division is 51.56790123, and the remainder is 3.7e-08.