10966 divided by 13 
We're looking to find how many times 13 goes into 10966.

Step 1:
If we divide 1 by 13, we get 0 and a remainder of 1.
The next digit of our result is 0. 
Result so far: 0.0
Subtract 0 from 1 to get 1.
Fetch the next digit (0) from the dividend, attach it to 1 and continue: 10 / 13

Step 2:
When dividing 10 by 13, we get 0 with a remainder of 10.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
If we take 0 away from 10, we end up with 10.
Bring next digit (9) of the dividend behind the 10 and repeat the process: 109 / 13

Let's proceed to step 3:
109 divided by 13 is 8 with a remainder of 5.
The next digit of our result is 8. 
Result so far: 8.0
Subtracting 104 from 109 leaves us with 5.
Append the next digit (6) from the dividend to 5 and continue with: 56 / 13

Advancing to step 4:
13 can be fit into 56 4 times, resulting in a remainder of 4.
Put 4 as the next digit of the answer. 
Result so far: 84.0
Deduct 52 from 56 and we're left with 4.
Append the next digit (6) from the dividend to 4 and continue with: 46 / 13

Advancing to step 5:
13 goes into 46 3 times with a remainder of 7.
Write down 3 as next digit of the result. 
Result so far: 843.0
Subtracting 39 from 46 leaves us with 7.
Since there are no more digits in the dividend, we add a zero to the remainder making it 70, and continue the process.

Step 6:
13 can be fit into 70 5 times, resulting in a remainder of 5.
Use 5 as the next digit of our solution. 
Result so far: 843.5
Subtracting 65 from 70 leaves us with 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

Moving on to step 7:
When dividing 50 by 13, we get 3 with a remainder of 11.
Write down 3 as next digit of the result. 
Result so far: 843.53
If we subtract 39 from 50, we get 11.
Since there are no more digits in the dividend, we add a zero to the remainder making it 110, and continue the process.

On to step 8:
When dividing 110 by 13, we get 8 with a remainder of 6.
Use 8 as the next digit of our solution. 
Result so far: 843.538
Subtract 104 from 110 to get 6.
Since there are no more digits in the dividend, we add a zero to the remainder making it 60, and continue the process.

Going ahead to step 9:
The number 13 fits into 60 4 times, leaving a remainder of 8.
Use 4 as the next digit of our solution. 
Result so far: 843.5384
The remainder is 8 after subtracting 52 from 60.
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 10:
80 divided by 13 is 6 with a remainder of 2.
Record the quotient 6 as the next digit in the result. 
Result so far: 843.53846
If we subtract 78 from 80, we get 2.
Since there are no more digits in the dividend, we add a zero to the remainder making it 20, and continue the process.

Step 11:
If we divide 20 by 13, we get 1 and a remainder of 7.
The next digit of our result is 1. 
Result so far: 843.538461
If we subtract 13 from 20, we get 7.
Since there are no more digits in the dividend, we add a zero to the remainder making it 70, and continue the process.

Moving on to step 12:
If we divide 70 by 13, we get 5 and a remainder of 5.
The number 5 becomes the next digit in our result. 
Result so far: 843.5384615
If we take 65 away from 70, we end up with 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

Let's proceed to step 13:
50 divided by 13 is 3 with a remainder of 11.
Put 3 as the next digit of the answer. 
Result so far: 843.53846153
Deduct 39 from 50 and we're left with 11.
Since there are no more digits in the dividend, we add a zero to the remainder making it 110, and continue the process.

After the division, we end up with 843.53846153 and a remainder of 1.1e-08.