510953 ÷ 13 
Our goal is to divide 510953 by 13.

Advancing to step 1:
If we divide 5 by 13, we get 0 and a remainder of 5.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
If we subtract 0 from 5, we get 5.
Grab the next digit (1) from the dividend, add it to 5, then carry on: 51 / 13

Moving on to step 2:
51 divided by 13 is 3 with a remainder of 12.
Use 3 as the next digit of our solution. 
Result so far: 3.0
If we take 39 away from 51, we end up with 12.
Grab the next digit (0) from the dividend, add it to 12, then carry on: 120 / 13

On to step 3:
When dividing 120 by 13, we get 9 with a remainder of 3.
The number 9 becomes the next digit in our result. 
Result so far: 39.0
Subtracting 117 from 120 leaves us with 3.
Include the next digit (9) from the dividend after 3, then repeat: 39 / 13

Moving on to step 4:
If we divide 39 by 13, we get 3 and a remainder of 0.
Write down 3 as next digit of the result. 
Result so far: 393.0
If we take 39 away from 39, we end up with 0.
Include the next digit (5) from the dividend after 0, then repeat: 5 / 13

Let's proceed to step 5:
5 divided by 13 is 0 with a remainder of 5.
The next digit of our result is 0. 
Result so far: 3930.0
Subtracting 0 from 5 leaves us with 5.
Grab the next digit (3) from the dividend, add it to 5, then carry on: 53 / 13

Let's proceed to step 6:
53 divided by 13 is 4 with a remainder of 1.
The next digit of our result is 4. 
Result so far: 39304.0
Subtract 52 from 53 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.

Going ahead to step 7:
10 divided by 13 is 0 with a remainder of 10.
Record the quotient 0 as the next digit in the result. 
Result so far: 39304.0
If we subtract 0 from 10, 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.

Let's proceed to step 8:
100 divided by 13 is 7 with a remainder of 9.
Use 7 as the next digit of our solution. 
Result so far: 39304.07
Deduct 91 from 100 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 13 is 6 with a remainder of 12.
Use 6 as the next digit of our solution. 
Result so far: 39304.076
If we take 78 away from 90, 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.

Step 10:
13 can be fit into 120 9 times, resulting in a remainder of 3.
Write down 9 as next digit of the result. 
Result so far: 39304.0769
Subtract 117 from 120 to get 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

Advancing to step 11:
The number 13 fits into 30 2 times, leaving a remainder of 4.
Record the quotient 2 as the next digit in the result. 
Result so far: 39304.07692
Deduct 26 from 30 and we're left with 4.
Since there are no more digits in the dividend, we add a zero to the remainder making it 40, and continue the process.

On to step 12:
When dividing 40 by 13, we get 3 with a remainder of 1.
The number 3 becomes the next digit in our result. 
Result so far: 39304.076923
Subtracting 39 from 40 leaves us 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.

Advancing to step 13:
When dividing 10 by 13, we get 0 with a remainder of 10.
Use 0 as the next digit of our solution. 
Result so far: 39304.076923
Subtract 0 from 10 to 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 14:
13 goes into 100 7 times with a remainder of 9.
Write down 7 as next digit of the result. 
Result so far: 39304.07692307
Subtracting 91 from 100 leaves us 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.

The final outcome is 39304.07692307, with a leftover of 9e-09.