10091601 ÷ 289 
The aim is to understand the frequency of 289 in 10091601.

Step 1:
When dividing 1 by 289, we get 0 with a remainder of 1.
Use 0 as the next digit of our solution. 
Result so far: 0.0
Deduct 0 from 1 and we're left with 1.
Include the next digit (0) from the dividend after 1, then repeat: 10 / 289

Advancing to step 2:
The number 289 fits into 10 0 times, leaving a remainder of 10.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
The remainder is 10 after subtracting 0 from 10.
Take the next digit (0) from the dividend and append it to 10, then repeat: 100 / 289

Going ahead to step 3:
289 goes into 100 0 times with a remainder of 100.
Use 0 as the next digit of our solution. 
Result so far: 0.0
Deduct 0 from 100 and we're left with 100.
Fetch the next digit (9) from the dividend, attach it to 100 and continue: 1009 / 289

Let's proceed to step 4:
289 can be fit into 1009 3 times, resulting in a remainder of 142.
The number 3 becomes the next digit in our result. 
Result so far: 3.0
Subtract 867 from 1009 to get 142.
Append the next digit (1) from the dividend to 142 and continue with: 1421 / 289

Step 5:
1421 divided by 289 is 4 with a remainder of 265.
Put 4 as the next digit of the answer. 
Result so far: 34.0
If we take 1156 away from 1421, we end up with 265.
Include the next digit (6) from the dividend after 265, then repeat: 2656 / 289

Let's proceed to step 6:
If we divide 2656 by 289, we get 9 and a remainder of 55.
The next digit of our result is 9. 
Result so far: 349.0
If we take 2601 away from 2656, we end up with 55.
Append the next digit (0) from the dividend to 55 and continue with: 550 / 289

Step 7:
289 goes into 550 1 times with a remainder of 261.
The next digit of our result is 1. 
Result so far: 3491.0
If we take 289 away from 550, we end up with 261.
Grab the next digit (1) from the dividend, add it to 261, then carry on: 2611 / 289

Step 8:
289 goes into 2611 9 times with a remainder of 10.
Write down 9 as next digit of the result. 
Result so far: 34919.0
If we take 2601 away from 2611, 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 9:
289 goes into 100 0 times with a remainder of 100.
The next digit of our result is 0. 
Result so far: 34919.0
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.

Let's proceed to step 10:
289 can be fit into 1000 3 times, resulting in a remainder of 133.
Write down 3 as next digit of the result. 
Result so far: 34919.03
If we subtract 867 from 1000, we get 133.
Since there are no more digits in the dividend, we add a zero to the remainder making it 1330, and continue the process.

Advancing to step 11:
1330 divided by 289 is 4 with a remainder of 174.
Write down 4 as next digit of the result. 
Result so far: 34919.034
If we take 1156 away from 1330, we end up with 174.
Since there are no more digits in the dividend, we add a zero to the remainder making it 1740, and continue the process.

On to step 12:
1740 divided by 289 is 6 with a remainder of 6.
Use 6 as the next digit of our solution. 
Result so far: 34919.0346
Subtract 1734 from 1740 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 13:
If we divide 60 by 289, we get 0 and a remainder of 60.
Use 0 as the next digit of our solution. 
Result so far: 34919.0346
If we subtract 0 from 60, we get 60.
Since there are no more digits in the dividend, we add a zero to the remainder making it 600, and continue the process.

Step 14:
289 goes into 600 2 times with a remainder of 22.
The number 2 becomes the next digit in our result. 
Result so far: 34919.034602
The remainder is 22 after subtracting 578 from 600.
Since there are no more digits in the dividend, we add a zero to the remainder making it 220, and continue the process.

Step 15:
The number 289 fits into 220 0 times, leaving a remainder of 220.
Record the quotient 0 as the next digit in the result. 
Result so far: 34919.034602
Subtracting 0 from 220 leaves us with 220.
Since there are no more digits in the dividend, we add a zero to the remainder making it 2200, and continue the process.

Step 16:
The number 289 fits into 2200 7 times, leaving a remainder of 177.
The next digit of our result is 7. 
Result so far: 34919.03460207
Subtracting 2023 from 2200 leaves us with 177.
Since there are no more digits in the dividend, we add a zero to the remainder making it 1770, and continue the process.

The quotient of the division is 34919.03460207, and the remainder is 1.77e-07.