We look at the division of 77643 by 71 
We're looking to find how many times 71 goes into 77643.

Moving on to step 1:
71 goes into 7 0 times with a remainder of 7.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
If we subtract 0 from 7, we get 7.
Grab the next digit (7) from the dividend, add it to 7, then carry on: 77 / 71

Advancing to step 2:
The number 71 fits into 77 1 times, leaving a remainder of 6.
Record the quotient 1 as the next digit in the result. 
Result so far: 1.0
Deduct 71 from 77 and we're left with 6.
Grab the next digit (6) from the dividend, add it to 6, then carry on: 66 / 71

Step 3:
66 divided by 71 is 0 with a remainder of 66.
The next digit of our result is 0. 
Result so far: 10.0
Subtracting 0 from 66 leaves us with 66.
Grab the next digit (4) from the dividend, add it to 66, then carry on: 664 / 71

On to step 4:
664 divided by 71 is 9 with a remainder of 25.
Record the quotient 9 as the next digit in the result. 
Result so far: 109.0
The remainder is 25 after subtracting 639 from 664.
Include the next digit (3) from the dividend after 25, then repeat: 253 / 71

Step 5:
71 can be fit into 253 3 times, resulting in a remainder of 40.
Record the quotient 3 as the next digit in the result. 
Result so far: 1093.0
The remainder is 40 after subtracting 213 from 253.
Since there are no more digits in the dividend, we add a zero to the remainder making it 400, and continue the process.

Step 6:
400 divided by 71 is 5 with a remainder of 45.
Put 5 as the next digit of the answer. 
Result so far: 1093.5
Subtracting 355 from 400 leaves us with 45.
Since there are no more digits in the dividend, we add a zero to the remainder making it 450, and continue the process.

Going ahead to step 7:
The number 71 fits into 450 6 times, leaving a remainder of 24.
Use 6 as the next digit of our solution. 
Result so far: 1093.56
If we take 426 away from 450, we end up with 24.
Since there are no more digits in the dividend, we add a zero to the remainder making it 240, and continue the process.

On to step 8:
If we divide 240 by 71, we get 3 and a remainder of 27.
Record the quotient 3 as the next digit in the result. 
Result so far: 1093.563
The remainder is 27 after subtracting 213 from 240.
Since there are no more digits in the dividend, we add a zero to the remainder making it 270, and continue the process.

Going ahead to step 9:
71 can be fit into 270 3 times, resulting in a remainder of 57.
Record the quotient 3 as the next digit in the result. 
Result so far: 1093.5633
If we subtract 213 from 270, we get 57.
Since there are no more digits in the dividend, we add a zero to the remainder making it 570, and continue the process.

Our division results in 1093.5633 with a remaining 0.00057.