102176 ÷ 13 
We want to figure out the number of times 102176 can be divided by 13.

On to step 1:
When dividing 1 by 13, we get 0 with a remainder of 1.
Write down 0 as next digit of the result. 
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 / 13

On to step 2:
10 divided by 13 is 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.
Fetch the next digit (2) from the dividend, attach it to 10 and continue: 102 / 13

Going ahead to step 3:
If we divide 102 by 13, we get 7 and a remainder of 11.
The number 7 becomes the next digit in our result. 
Result so far: 7.0
The remainder is 11 after subtracting 91 from 102.
Append the next digit (1) from the dividend to 11 and continue with: 111 / 13

Advancing to step 4:
13 goes into 111 8 times with a remainder of 7.
Record the quotient 8 as the next digit in the result. 
Result so far: 78.0
The remainder is 7 after subtracting 104 from 111.
Include the next digit (7) from the dividend after 7, then repeat: 77 / 13

Moving on to step 5:
If we divide 77 by 13, we get 5 and a remainder of 12.
Record the quotient 5 as the next digit in the result. 
Result so far: 785.0
Deduct 65 from 77 and we're left with 12.
Include the next digit (6) from the dividend after 12, then repeat: 126 / 13

Moving on to step 6:
When dividing 126 by 13, we get 9 with a remainder of 9.
Use 9 as the next digit of our solution. 
Result so far: 7859.0
Deduct 117 from 126 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.

Advancing to step 7:
13 goes into 90 6 times with a remainder of 12.
Record the quotient 6 as the next digit in the result. 
Result so far: 7859.6
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.

Let's proceed to step 8:
If we divide 120 by 13, we get 9 and a remainder of 3.
Put 9 as the next digit of the answer. 
Result so far: 7859.69
Subtracting 117 from 120 leaves us with 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

Step 9:
30 divided by 13 is 2 with a remainder of 4.
The next digit of our result is 2. 
Result so far: 7859.692
Deduct 26 from 30 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.

Advancing to step 10:
40 divided by 13 is 3 with a remainder of 1.
Use 3 as the next digit of our solution. 
Result so far: 7859.6923
If we subtract 39 from 40, we 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.

Step 11:
When dividing 10 by 13, we get 0 with a remainder of 10.
The number 0 becomes the next digit in our result. 
Result so far: 7859.6923
If we take 0 away from 10, we end up 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.

Moving on to step 12:
The number 13 fits into 100 7 times, leaving a remainder of 9.
Put 7 as the next digit of the answer. 
Result so far: 7859.692307
Subtract 91 from 100 to 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.

Moving on to step 13:
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: 7859.6923076
Deduct 78 from 90 and we're left 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.

Step 14:
120 divided by 13 is 9 with a remainder of 3.
Write down 9 as next digit of the result. 
Result so far: 7859.69230769
Subtract 117 from 120 to 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.

Our division results in 7859.69230769 with a remaining 3e-09.