We look at the division of 581177 by 45 
We're looking to find how many times 45 goes into 581177.

Let's proceed to step 1:
5 divided by 45 is 0 with a remainder of 5.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
Deduct 0 from 5 and we're left with 5.
Take the next digit (8) from the dividend and append it to 5, then repeat: 58 / 45

Let's proceed to step 2:
When dividing 58 by 45, we get 1 with a remainder of 13.
The number 1 becomes the next digit in our result. 
Result so far: 1.0
If we take 45 away from 58, we end up with 13.
Include the next digit (1) from the dividend after 13, then repeat: 131 / 45

Let's proceed to step 3:
131 divided by 45 is 2 with a remainder of 41.
Write down 2 as next digit of the result. 
Result so far: 12.0
If we subtract 90 from 131, we get 41.
Grab the next digit (1) from the dividend, add it to 41, then carry on: 411 / 45

Let's proceed to step 4:
If we divide 411 by 45, we get 9 and a remainder of 6.
Record the quotient 9 as the next digit in the result. 
Result so far: 129.0
Subtract 405 from 411 to get 6.
Bring next digit (7) of the dividend behind the 6 and repeat the process: 67 / 45

Step 5:
45 can be fit into 67 1 times, resulting in a remainder of 22.
Write down 1 as next digit of the result. 
Result so far: 1291.0
The remainder is 22 after subtracting 45 from 67.
Fetch the next digit (7) from the dividend, attach it to 22 and continue: 227 / 45

Advancing to step 6:
When dividing 227 by 45, we get 5 with a remainder of 2.
Record the quotient 5 as the next digit in the result. 
Result so far: 12915.0
If we take 225 away from 227, we end up with 2.
Since there are no more digits in the dividend, we add a zero to the remainder making it 20, and continue the process.

Step 7:
45 goes into 20 0 times with a remainder of 20.
Use 0 as the next digit of our solution. 
Result so far: 12915.0
Subtracting 0 from 20 leaves us with 20.
Since there are no more digits in the dividend, we add a zero to the remainder making it 200, and continue the process.

Let's proceed to step 8:
45 goes into 200 4 times with a remainder of 20.
The next digit of our result is 4. 
Result so far: 12915.04
The remainder is 20 after subtracting 180 from 200.
Since there are no more digits in the dividend, we add a zero to the remainder making it 200, and continue the process.

Step 9:
The number 45 fits into 200 4 times, leaving a remainder of 20.
The number 4 becomes the next digit in our result. 
Result so far: 12915.044
The remainder is 20 after subtracting 180 from 200.
Since there are no more digits in the dividend, we add a zero to the remainder making it 200, and continue the process.

After the division, we end up with 12915.044 and a remainder of 0.002.