We look at the division of 105781 by 9 
We're looking to find how many times 9 goes into 105781.

Going ahead to step 1:
The number 9 fits into 1 0 times, leaving a remainder of 1.
Put 0 as the next digit of the answer. 
Result so far: 0.0
The remainder is 1 after subtracting 0 from 1.
Append the next digit (0) from the dividend to 1 and continue with: 10 / 9

Moving on to step 2:
9 can be fit into 10 1 times, resulting in a remainder of 1.
Record the quotient 1 as the next digit in the result. 
Result so far: 1.0
If we subtract 9 from 10, we get 1.
Take the next digit (5) from the dividend and append it to 1, then repeat: 15 / 9

Going ahead to step 3:
15 divided by 9 is 1 with a remainder of 6.
Write down 1 as next digit of the result. 
Result so far: 11.0
The remainder is 6 after subtracting 9 from 15.
Include the next digit (7) from the dividend after 6, then repeat: 67 / 9

Going ahead to step 4:
The number 9 fits into 67 7 times, leaving a remainder of 4.
Use 7 as the next digit of our solution. 
Result so far: 117.0
The remainder is 4 after subtracting 63 from 67.
Append the next digit (8) from the dividend to 4 and continue with: 48 / 9

Let's proceed to step 5:
9 can be fit into 48 5 times, resulting in a remainder of 3.
Record the quotient 5 as the next digit in the result. 
Result so far: 1175.0
Subtract 45 from 48 to get 3.
Take the next digit (1) from the dividend and append it to 3, then repeat: 31 / 9

Advancing to step 6:
The number 9 fits into 31 3 times, leaving a remainder of 4.
The next digit of our result is 3. 
Result so far: 11753.0
Deduct 27 from 31 and we're left with 4.
Since there are no more digits in the dividend, we add a zero to the remainder making it 40, and continue the process.

On to step 7:
9 goes into 40 4 times with a remainder of 4.
The next digit of our result is 4. 
Result so far: 11753.4
Deduct 36 from 40 and we're left with 4.
Since there are no more digits in the dividend, we add a zero to the remainder making it 40, and continue the process.

Let's proceed to step 8:
9 goes into 40 4 times with a remainder of 4.
Record the quotient 4 as the next digit in the result. 
Result so far: 11753.44
If we subtract 36 from 40, we get 4.
Since there are no more digits in the dividend, we add a zero to the remainder making it 40, and continue the process.

After the division, we end up with 11753.44 and a remainder of 0.004.