We look at the division of 338269 by 3 
We're looking to find how many times 3 goes into 338269.

Step 1:
3 can be fit into 3 1 times, resulting in a remainder of 0.
Use 1 as the next digit of our solution. 
Result so far: 1.0
Subtract 3 from 3 to get 0.
Fetch the next digit (3) from the dividend, attach it to 0 and continue: 3 / 3

Going ahead to step 2:
3 can be fit into 3 1 times, resulting in a remainder of 0.
Write down 1 as next digit of the result. 
Result so far: 11.0
If we subtract 3 from 3, we get 0.
Include the next digit (8) from the dividend after 0, then repeat: 8 / 3

Moving on to step 3:
When dividing 8 by 3, we get 2 with a remainder of 2.
Put 2 as the next digit of the answer. 
Result so far: 112.0
The remainder is 2 after subtracting 6 from 8.
Take the next digit (2) from the dividend and append it to 2, then repeat: 22 / 3

Let's proceed to step 4:
The number 3 fits into 22 7 times, leaving a remainder of 1.
Write down 7 as next digit of the result. 
Result so far: 1127.0
If we subtract 21 from 22, we get 1.
Take the next digit (6) from the dividend and append it to 1, then repeat: 16 / 3

Going ahead to step 5:
When dividing 16 by 3, we get 5 with a remainder of 1.
Use 5 as the next digit of our solution. 
Result so far: 11275.0
If we take 15 away from 16, we end up with 1.
Include the next digit (9) from the dividend after 1, then repeat: 19 / 3

On to step 6:
If we divide 19 by 3, we get 6 and a remainder of 1.
Record the quotient 6 as the next digit in the result. 
Result so far: 112756.0
Deduct 18 from 19 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.

Step 7:
When dividing 10 by 3, we get 3 with a remainder of 1.
The next digit of our result is 3. 
Result so far: 112756.3
Subtract 9 from 10 to 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.

On to step 8:
10 divided by 3 is 3 with a remainder of 1.
Put 3 as the next digit of the answer. 
Result so far: 112756.33
Subtract 9 from 10 to 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.

After the division, we end up with 112756.33 and a remainder of 0.001.