We look at the division of 259113 by 13 
We're looking to find how many times 13 goes into 259113.

On to step 1:
13 goes into 2 0 times with a remainder of 2.
The number 0 becomes the next digit in our result. 
Result so far: 0.0
The remainder is 2 after subtracting 0 from 2.
Grab the next digit (5) from the dividend, add it to 2, then carry on: 25 / 13

On to step 2:
25 divided by 13 is 1 with a remainder of 12.
Use 1 as the next digit of our solution. 
Result so far: 1.0
The remainder is 12 after subtracting 13 from 25.
Grab the next digit (9) from the dividend, add it to 12, then carry on: 129 / 13

Advancing to step 3:
129 divided by 13 is 9 with a remainder of 12.
Put 9 as the next digit of the answer. 
Result so far: 19.0
The remainder is 12 after subtracting 117 from 129.
Append the next digit (1) from the dividend to 12 and continue with: 121 / 13

Moving on to step 4:
121 divided by 13 is 9 with a remainder of 4.
Use 9 as the next digit of our solution. 
Result so far: 199.0
The remainder is 4 after subtracting 117 from 121.
Append the next digit (1) from the dividend to 4 and continue with: 41 / 13

Step 5:
If we divide 41 by 13, we get 3 and a remainder of 2.
Write down 3 as next digit of the result. 
Result so far: 1993.0
Subtracting 39 from 41 leaves us with 2.
Bring next digit (3) of the dividend behind the 2 and repeat the process: 23 / 13

Moving on to step 6:
13 goes into 23 1 times with a remainder of 10.
The next digit of our result is 1. 
Result so far: 19931.0
Deduct 13 from 23 and we're left 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.

Going ahead to step 7:
100 divided by 13 is 7 with a remainder of 9.
The next digit of our result is 7. 
Result so far: 19931.7
If we subtract 91 from 100, we get 9.
Since there are no more digits in the dividend, we add a zero to the remainder making it 90, and continue the process.

Going ahead to step 8:
13 can be fit into 90 6 times, resulting in a remainder of 12.
Use 6 as the next digit of our solution. 
Result so far: 19931.76
Subtracting 78 from 90 leaves us with 12.
Since there are no more digits in the dividend, we add a zero to the remainder making it 120, and continue the process.

On to step 9:
120 divided by 13 is 9 with a remainder of 3.
Put 9 as the next digit of the answer. 
Result so far: 19931.769
If we subtract 117 from 120, we get 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

Advancing to step 10:
When dividing 30 by 13, we get 2 with a remainder of 4.
Use 2 as the next digit of our solution. 
Result so far: 19931.7692
Subtracting 26 from 30 leaves us 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 11:
When dividing 40 by 13, we get 3 with a remainder of 1.
The number 3 becomes the next digit in our result. 
Result so far: 19931.76923
Subtract 39 from 40 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 12:
If we divide 10 by 13, we get 0 and a remainder of 10.
Write down 0 as next digit of the result. 
Result so far: 19931.76923
If we subtract 0 from 10, we get 10.
Since there are no more digits in the dividend, we add a zero to the remainder making it 100, and continue the process.

Step 13:
13 can be fit into 100 7 times, resulting in a remainder of 9.
Use 7 as the next digit of our solution. 
Result so far: 19931.7692307
Deduct 91 from 100 and we're left 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.

Step 14:
13 can be fit into 90 6 times, resulting in a remainder of 12.
The next digit of our result is 6. 
Result so far: 19931.76923076
The remainder is 12 after subtracting 78 from 90.
Since there are no more digits in the dividend, we add a zero to the remainder making it 120, and continue the process.

After the division, we end up with 19931.76923076 and a remainder of 1.2e-08.