6577501 ÷ 431 
The aim is to understand the frequency of 431 in 6577501.

Advancing to step 1:
The number 431 fits into 6 0 times, leaving a remainder of 6.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
The remainder is 6 after subtracting 0 from 6.
Append the next digit (5) from the dividend to 6 and continue with: 65 / 431

On to step 2:
If we divide 65 by 431, we get 0 and a remainder of 65.
Write down 0 as next digit of the result. 
Result so far: 0.0
If we take 0 away from 65, we end up with 65.
Append the next digit (7) from the dividend to 65 and continue with: 657 / 431

Advancing to step 3:
431 goes into 657 1 times with a remainder of 226.
The next digit of our result is 1. 
Result so far: 1.0
Deduct 431 from 657 and we're left with 226.
Grab the next digit (7) from the dividend, add it to 226, then carry on: 2267 / 431

On to step 4:
431 goes into 2267 5 times with a remainder of 112.
Record the quotient 5 as the next digit in the result. 
Result so far: 15.0
Deduct 2155 from 2267 and we're left with 112.
Include the next digit (5) from the dividend after 112, then repeat: 1125 / 431

Moving on to step 5:
431 goes into 1125 2 times with a remainder of 263.
Record the quotient 2 as the next digit in the result. 
Result so far: 152.0
Subtracting 862 from 1125 leaves us with 263.
Fetch the next digit (0) from the dividend, attach it to 263 and continue: 2630 / 431

Advancing to step 6:
2630 divided by 431 is 6 with a remainder of 44.
Use 6 as the next digit of our solution. 
Result so far: 1526.0
The remainder is 44 after subtracting 2586 from 2630.
Take the next digit (1) from the dividend and append it to 44, then repeat: 441 / 431

Step 7:
The number 431 fits into 441 1 times, leaving a remainder of 10.
The next digit of our result is 1. 
Result so far: 15261.0
If we subtract 431 from 441, 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 8:
431 goes into 100 0 times with a remainder of 100.
Write down 0 as next digit of the result. 
Result so far: 15261.0
The remainder is 100 after subtracting 0 from 100.
Since there are no more digits in the dividend, we add a zero to the remainder making it 1000, and continue the process.

On to step 9:
1000 divided by 431 is 2 with a remainder of 138.
The next digit of our result is 2. 
Result so far: 15261.02
Subtract 862 from 1000 to get 138.
Since there are no more digits in the dividend, we add a zero to the remainder making it 1380, and continue the process.

On to step 10:
When dividing 1380 by 431, we get 3 with a remainder of 87.
The number 3 becomes the next digit in our result. 
Result so far: 15261.023
If we subtract 1293 from 1380, we get 87.
Since there are no more digits in the dividend, we add a zero to the remainder making it 870, and continue the process.

Step 11:
431 can be fit into 870 2 times, resulting in a remainder of 8.
The number 2 becomes the next digit in our result. 
Result so far: 15261.0232
Subtracting 862 from 870 leaves us with 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Advancing to step 12:
When dividing 80 by 431, we get 0 with a remainder of 80.
Use 0 as the next digit of our solution. 
Result so far: 15261.0232
The remainder is 80 after subtracting 0 from 80.
Since there are no more digits in the dividend, we add a zero to the remainder making it 800, and continue the process.

Going ahead to step 13:
When dividing 800 by 431, we get 1 with a remainder of 369.
Record the quotient 1 as the next digit in the result. 
Result so far: 15261.023201
If we take 431 away from 800, we end up with 369.
Since there are no more digits in the dividend, we add a zero to the remainder making it 3690, and continue the process.

Step 14:
The number 431 fits into 3690 8 times, leaving a remainder of 242.
Record the quotient 8 as the next digit in the result. 
Result so far: 15261.0232018
Deduct 3448 from 3690 and we're left with 242.
Since there are no more digits in the dividend, we add a zero to the remainder making it 2420, and continue the process.

Moving on to step 15:
431 goes into 2420 5 times with a remainder of 265.
Put 5 as the next digit of the answer. 
Result so far: 15261.02320185
Subtract 2155 from 2420 to get 265.
Since there are no more digits in the dividend, we add a zero to the remainder making it 2650, and continue the process.

After the division, we end up with 15261.02320185 and a remainder of 2.65e-07.