43165 ÷ 13 
Our goal is to divide 43165 by 13.

Advancing to step 1:
If we divide 4 by 13, we get 0 and a remainder of 4.
Write down 0 as next digit of the result. 
Result so far: 0.0
If we take 0 away from 4, we end up with 4.
Bring next digit (3) of the dividend behind the 4 and repeat the process: 43 / 13

Let's proceed to step 2:
If we divide 43 by 13, we get 3 and a remainder of 4.
The number 3 becomes the next digit in our result. 
Result so far: 3.0
If we subtract 39 from 43, we get 4.
Fetch the next digit (1) from the dividend, attach it to 4 and continue: 41 / 13

Moving on to step 3:
41 divided by 13 is 3 with a remainder of 2.
Record the quotient 3 as the next digit in the result. 
Result so far: 33.0
If we subtract 39 from 41, we get 2.
Append the next digit (6) from the dividend to 2 and continue with: 26 / 13

Let's proceed to step 4:
13 goes into 26 2 times with a remainder of 0.
Put 2 as the next digit of the answer. 
Result so far: 332.0
Subtract 26 from 26 to get 0.
Grab the next digit (5) from the dividend, add it to 0, then carry on: 5 / 13

Moving on to step 5:
13 can be fit into 5 0 times, resulting in a remainder of 5.
Use 0 as the next digit of our solution. 
Result so far: 3320.0
If we take 0 away from 5, we end up with 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

On to step 6:
The number 13 fits into 50 3 times, leaving a remainder of 11.
Record the quotient 3 as the next digit in the result. 
Result so far: 3320.3
If we take 39 away from 50, we end up with 11.
Since there are no more digits in the dividend, we add a zero to the remainder making it 110, and continue the process.

On to step 7:
110 divided by 13 is 8 with a remainder of 6.
Record the quotient 8 as the next digit in the result. 
Result so far: 3320.38
If we take 104 away from 110, we end up with 6.
Since there are no more digits in the dividend, we add a zero to the remainder making it 60, and continue the process.

Advancing to step 8:
13 goes into 60 4 times with a remainder of 8.
The number 4 becomes the next digit in our result. 
Result so far: 3320.384
Subtract 52 from 60 to get 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Moving on to step 9:
13 goes into 80 6 times with a remainder of 2.
Put 6 as the next digit of the answer. 
Result so far: 3320.3846
The remainder is 2 after subtracting 78 from 80.
Since there are no more digits in the dividend, we add a zero to the remainder making it 20, and continue the process.

Step 10:
When dividing 20 by 13, we get 1 with a remainder of 7.
Record the quotient 1 as the next digit in the result. 
Result so far: 3320.38461
Subtract 13 from 20 to get 7.
Since there are no more digits in the dividend, we add a zero to the remainder making it 70, and continue the process.

On to step 11:
If we divide 70 by 13, we get 5 and a remainder of 5.
The next digit of our result is 5. 
Result so far: 3320.384615
Deduct 65 from 70 and we're left with 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

Step 12:
13 goes into 50 3 times with a remainder of 11.
The next digit of our result is 3. 
Result so far: 3320.3846153
If we take 39 away from 50, we end up with 11.
Since there are no more digits in the dividend, we add a zero to the remainder making it 110, and continue the process.

Going ahead to step 13:
110 divided by 13 is 8 with a remainder of 6.
The number 8 becomes the next digit in our result. 
Result so far: 3320.38461538
Subtract 104 from 110 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.

In conclusion, the division gives 3320.38461538 with a balance of 6e-09.