We're dividing 684358 by 13 
We want to figure out the number of times 684358 can be divided by 13.

Moving on to step 1:
If we divide 6 by 13, we get 0 and a remainder of 6.
Write down 0 as next digit of the result. 
Result so far: 0.0
Subtract 0 from 6 to get 6.
Bring next digit (8) of the dividend behind the 6 and repeat the process: 68 / 13

Moving on to step 2:
The number 13 fits into 68 5 times, leaving a remainder of 3.
Put 5 as the next digit of the answer. 
Result so far: 5.0
Subtract 65 from 68 to get 3.
Grab the next digit (4) from the dividend, add it to 3, then carry on: 34 / 13

Step 3:
If we divide 34 by 13, we get 2 and a remainder of 8.
Write down 2 as next digit of the result. 
Result so far: 52.0
Subtract 26 from 34 to get 8.
Take the next digit (3) from the dividend and append it to 8, then repeat: 83 / 13

Going ahead to step 4:
13 goes into 83 6 times with a remainder of 5.
Use 6 as the next digit of our solution. 
Result so far: 526.0
Subtract 78 from 83 to get 5.
Fetch the next digit (5) from the dividend, attach it to 5 and continue: 55 / 13

Step 5:
13 can be fit into 55 4 times, resulting in a remainder of 3.
Use 4 as the next digit of our solution. 
Result so far: 5264.0
If we take 52 away from 55, we end up with 3.
Take the next digit (8) from the dividend and append it to 3, then repeat: 38 / 13

Moving on to step 6:
When dividing 38 by 13, we get 2 with a remainder of 12.
Use 2 as the next digit of our solution. 
Result so far: 52642.0
Deduct 26 from 38 and we're left 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.

Advancing to step 7:
13 can be fit into 120 9 times, resulting in a remainder of 3.
Put 9 as the next digit of the answer. 
Result so far: 52642.9
The remainder is 3 after subtracting 117 from 120.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

On to step 8:
13 goes into 30 2 times with a remainder of 4.
The number 2 becomes the next digit in our result. 
Result so far: 52642.92
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.

Step 9:
40 divided by 13 is 3 with a remainder of 1.
Use 3 as the next digit of our solution. 
Result so far: 52642.923
Subtract 39 from 40 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.

Advancing to step 10:
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: 52642.923
Deduct 0 from 10 and we're left 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.

Step 11:
If we divide 100 by 13, we get 7 and a remainder of 9.
Record the quotient 7 as the next digit in the result. 
Result so far: 52642.92307
If we subtract 91 from 100, we get 9.
Since there are no more digits in the dividend, we add a zero to the remainder making it 90, and continue the process.

On to step 12:
13 can be fit into 90 6 times, resulting in a remainder of 12.
Put 6 as the next digit of the answer. 
Result so far: 52642.923076
Subtract 78 from 90 to 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.

Advancing to step 13:
120 divided by 13 is 9 with a remainder of 3.
The number 9 becomes the next digit in our result. 
Result so far: 52642.9230769
If we subtract 117 from 120, we 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.

Step 14:
If we divide 30 by 13, we get 2 and a remainder of 4.
The next digit of our result is 2. 
Result so far: 52642.92307692
If we subtract 26 from 30, we get 4.
Since there are no more digits in the dividend, we add a zero to the remainder making it 40, and continue the process.

Our division results in 52642.92307692 with a remaining 4e-09.