201566 ÷ 13 
The aim is to understand the frequency of 13 in 201566.

Let's proceed to step 1:
When dividing 2 by 13, we get 0 with a remainder of 2.
The number 0 becomes the next digit in our result. 
Result so far: 0.0
Subtracting 0 from 2 leaves us with 2.
Include the next digit (0) from the dividend after 2, then repeat: 20 / 13

Going ahead to step 2:
13 goes into 20 1 times with a remainder of 7.
The number 1 becomes the next digit in our result. 
Result so far: 1.0
Deduct 13 from 20 and we're left with 7.
Fetch the next digit (1) from the dividend, attach it to 7 and continue: 71 / 13

On to step 3:
The number 13 fits into 71 5 times, leaving a remainder of 6.
Use 5 as the next digit of our solution. 
Result so far: 15.0
If we subtract 65 from 71, we get 6.
Include the next digit (5) from the dividend after 6, then repeat: 65 / 13

Step 4:
The number 13 fits into 65 5 times, leaving a remainder of 0.
Put 5 as the next digit of the answer. 
Result so far: 155.0
Subtracting 65 from 65 leaves us with 0.
Append the next digit (6) from the dividend to 0 and continue with: 6 / 13

Let's proceed to step 5:
13 can be fit into 6 0 times, resulting in a remainder of 6.
The next digit of our result is 0. 
Result so far: 1550.0
Deduct 0 from 6 and we're left with 6.
Take the next digit (6) from the dividend and append it to 6, then repeat: 66 / 13

Advancing to step 6:
13 can be fit into 66 5 times, resulting in a remainder of 1.
The number 5 becomes the next digit in our result. 
Result so far: 15505.0
Subtracting 65 from 66 leaves us with 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 7:
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: 15505.0
Subtracting 0 from 10 leaves us 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.

On to step 8:
When dividing 100 by 13, we get 7 with a remainder of 9.
Put 7 as the next digit of the answer. 
Result so far: 15505.07
Subtracting 91 from 100 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 9:
When dividing 90 by 13, we get 6 with a remainder of 12.
Write down 6 as next digit of the result. 
Result so far: 15505.076
If we subtract 78 from 90, we get 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 10:
The number 13 fits into 120 9 times, leaving a remainder of 3.
The number 9 becomes the next digit in our result. 
Result so far: 15505.0769
Deduct 117 from 120 and we're left 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.

Advancing to step 11:
13 goes into 30 2 times with a remainder of 4.
Use 2 as the next digit of our solution. 
Result so far: 15505.07692
The remainder is 4 after subtracting 26 from 30.
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 12:
When dividing 40 by 13, we get 3 with a remainder of 1.
The next digit of our result is 3. 
Result so far: 15505.076923
Deduct 39 from 40 and we're left with 1.
Since there are no more digits in the dividend, we add a zero to the remainder making it 10, and continue the process.

Let's proceed to step 13:
13 goes into 10 0 times with a remainder of 10.
Write down 0 as next digit of the result. 
Result so far: 15505.076923
Subtracting 0 from 10 leaves us 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 14:
If we divide 100 by 13, we get 7 and a remainder of 9.
Use 7 as the next digit of our solution. 
Result so far: 15505.07692307
Subtracting 91 from 100 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.

The final outcome is 15505.07692307, with a leftover of 9e-09.