105259 ÷ 71 
We want to figure out the number of times 105259 can be divided by 71.

Moving on to step 1:
When dividing 1 by 71, we get 0 with a remainder of 1.
Use 0 as the next digit of our solution. 
Result so far: 0.0
The remainder is 1 after subtracting 0 from 1.
Grab the next digit (0) from the dividend, add it to 1, then carry on: 10 / 71

Going ahead to step 2:
If we divide 10 by 71, we get 0 and a remainder of 10.
The number 0 becomes the next digit in our result. 
Result so far: 0.0
If we subtract 0 from 10, we get 10.
Grab the next digit (5) from the dividend, add it to 10, then carry on: 105 / 71

On to step 3:
The number 71 fits into 105 1 times, leaving a remainder of 34.
The number 1 becomes the next digit in our result. 
Result so far: 1.0
If we subtract 71 from 105, we get 34.
Append the next digit (2) from the dividend to 34 and continue with: 342 / 71

Moving on to step 4:
If we divide 342 by 71, we get 4 and a remainder of 58.
The number 4 becomes the next digit in our result. 
Result so far: 14.0
If we subtract 284 from 342, we get 58.
Take the next digit (5) from the dividend and append it to 58, then repeat: 585 / 71

On to step 5:
If we divide 585 by 71, we get 8 and a remainder of 17.
Write down 8 as next digit of the result. 
Result so far: 148.0
The remainder is 17 after subtracting 568 from 585.
Take the next digit (9) from the dividend and append it to 17, then repeat: 179 / 71

Moving on to step 6:
If we divide 179 by 71, we get 2 and a remainder of 37.
The next digit of our result is 2. 
Result so far: 1482.0
The remainder is 37 after subtracting 142 from 179.
Since there are no more digits in the dividend, we add a zero to the remainder making it 370, and continue the process.

Step 7:
The number 71 fits into 370 5 times, leaving a remainder of 15.
Put 5 as the next digit of the answer. 
Result so far: 1482.5
If we take 355 away from 370, we end up with 15.
Since there are no more digits in the dividend, we add a zero to the remainder making it 150, and continue the process.

On to step 8:
If we divide 150 by 71, we get 2 and a remainder of 8.
Record the quotient 2 as the next digit in the result. 
Result so far: 1482.52
Subtract 142 from 150 to get 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Let's proceed to step 9:
80 divided by 71 is 1 with a remainder of 9.
The next digit of our result is 1. 
Result so far: 1482.521
Subtracting 71 from 80 leaves us with 9.
Since there are no more digits in the dividend, we add a zero to the remainder making it 90, and continue the process.

Let's proceed to step 10:
If we divide 90 by 71, we get 1 and a remainder of 19.
The number 1 becomes the next digit in our result. 
Result so far: 1482.5211
Subtracting 71 from 90 leaves us with 19.
Since there are no more digits in the dividend, we add a zero to the remainder making it 190, and continue the process.

The final result is 1482.5211 with a remainder of 0.00019.