338566 ÷ 37 
We're looking to find how many times 37 goes into 338566.

Advancing to step 1:
3 divided by 37 is 0 with a remainder of 3.
Use 0 as the next digit of our solution. 
Result so far: 0.0
Subtracting 0 from 3 leaves us with 3.
Take the next digit (3) from the dividend and append it to 3, then repeat: 33 / 37

Advancing to step 2:
33 divided by 37 is 0 with a remainder of 33.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
Subtract 0 from 33 to get 33.
Grab the next digit (8) from the dividend, add it to 33, then carry on: 338 / 37

Let's proceed to step 3:
338 divided by 37 is 9 with a remainder of 5.
Record the quotient 9 as the next digit in the result. 
Result so far: 9.0
If we take 333 away from 338, we end up with 5.
Append the next digit (5) from the dividend to 5 and continue with: 55 / 37

Moving on to step 4:
When dividing 55 by 37, we get 1 with a remainder of 18.
Write down 1 as next digit of the result. 
Result so far: 91.0
Deduct 37 from 55 and we're left with 18.
Grab the next digit (6) from the dividend, add it to 18, then carry on: 186 / 37

Step 5:
The number 37 fits into 186 5 times, leaving a remainder of 1.
The number 5 becomes the next digit in our result. 
Result so far: 915.0
Deduct 185 from 186 and we're left with 1.
Append the next digit (6) from the dividend to 1 and continue with: 16 / 37

Advancing to step 6:
37 goes into 16 0 times with a remainder of 16.
The number 0 becomes the next digit in our result. 
Result so far: 9150.0
Subtract 0 from 16 to get 16.
Since there are no more digits in the dividend, we add a zero to the remainder making it 160, and continue the process.

On to step 7:
If we divide 160 by 37, we get 4 and a remainder of 12.
Put 4 as the next digit of the answer. 
Result so far: 9150.4
Subtracting 148 from 160 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.

Going ahead to step 8:
120 divided by 37 is 3 with a remainder of 9.
Record the quotient 3 as the next digit in the result. 
Result so far: 9150.43
Deduct 111 from 120 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 9:
90 divided by 37 is 2 with a remainder of 16.
The number 2 becomes the next digit in our result. 
Result so far: 9150.432
If we take 74 away from 90, we end up with 16.
Since there are no more digits in the dividend, we add a zero to the remainder making it 160, and continue the process.

Going ahead to step 10:
The number 37 fits into 160 4 times, leaving a remainder of 12.
Write down 4 as next digit of the result. 
Result so far: 9150.4324
If we subtract 148 from 160, 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.

Moving on to step 11:
When dividing 120 by 37, we get 3 with a remainder of 9.
Record the quotient 3 as the next digit in the result. 
Result so far: 9150.43243
If we take 111 away from 120, we end up 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 12:
When dividing 90 by 37, we get 2 with a remainder of 16.
The next digit of our result is 2. 
Result so far: 9150.432432
Subtracting 74 from 90 leaves us with 16.
Since there are no more digits in the dividend, we add a zero to the remainder making it 160, and continue the process.

Moving on to step 13:
When dividing 160 by 37, we get 4 with a remainder of 12.
The number 4 becomes the next digit in our result. 
Result so far: 9150.4324324
If we take 148 away from 160, we end up 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.

Going ahead to step 14:
The number 37 fits into 120 3 times, leaving a remainder of 9.
Put 3 as the next digit of the answer. 
Result so far: 9150.43243243
Deduct 111 from 120 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.

In conclusion, the division gives 9150.43243243 with a balance of 9e-09.