586603 ÷ 79 
Our goal is to divide 586603 by 79.

Going ahead to step 1:
If we divide 5 by 79, 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 take 0 away from 5, we end up with 5.
Append the next digit (8) from the dividend to 5 and continue with: 58 / 79

Advancing to step 2:
79 goes into 58 0 times with a remainder of 58.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
If we take 0 away from 58, we end up with 58.
Take the next digit (6) from the dividend and append it to 58, then repeat: 586 / 79

Let's proceed to step 3:
79 goes into 586 7 times with a remainder of 33.
Use 7 as the next digit of our solution. 
Result so far: 7.0
If we subtract 553 from 586, we get 33.
Include the next digit (6) from the dividend after 33, then repeat: 336 / 79

Advancing to step 4:
79 goes into 336 4 times with a remainder of 20.
The next digit of our result is 4. 
Result so far: 74.0
Subtracting 316 from 336 leaves us with 20.
Fetch the next digit (0) from the dividend, attach it to 20 and continue: 200 / 79

Let's proceed to step 5:
79 goes into 200 2 times with a remainder of 42.
The next digit of our result is 2. 
Result so far: 742.0
Deduct 158 from 200 and we're left with 42.
Bring next digit (3) of the dividend behind the 42 and repeat the process: 423 / 79

Moving on to step 6:
If we divide 423 by 79, we get 5 and a remainder of 28.
The next digit of our result is 5. 
Result so far: 7425.0
If we take 395 away from 423, we end up with 28.
Since there are no more digits in the dividend, we add a zero to the remainder making it 280, and continue the process.

Going ahead to step 7:
If we divide 280 by 79, we get 3 and a remainder of 43.
The next digit of our result is 3. 
Result so far: 7425.3
Subtract 237 from 280 to get 43.
Since there are no more digits in the dividend, we add a zero to the remainder making it 430, and continue the process.

Advancing to step 8:
When dividing 430 by 79, we get 5 with a remainder of 35.
Put 5 as the next digit of the answer. 
Result so far: 7425.35
Subtracting 395 from 430 leaves us with 35.
Since there are no more digits in the dividend, we add a zero to the remainder making it 350, and continue the process.

Advancing to step 9:
350 divided by 79 is 4 with a remainder of 34.
The number 4 becomes the next digit in our result. 
Result so far: 7425.354
Deduct 316 from 350 and we're left with 34.
Since there are no more digits in the dividend, we add a zero to the remainder making it 340, and continue the process.

On to step 10:
340 divided by 79 is 4 with a remainder of 24.
Write down 4 as next digit of the result. 
Result so far: 7425.3544
Deduct 316 from 340 and we're left with 24.
Since there are no more digits in the dividend, we add a zero to the remainder making it 240, and continue the process.

Our division results in 7425.3544 with a remaining 0.00024.