We look at the division of 25583 by 45 
We want to divide 25583 by 45.

On to step 1:
The number 45 fits into 2 0 times, leaving a remainder of 2.
Use 0 as the next digit of our solution. 
Result so far: 0.0
If we take 0 away from 2, we end up with 2.
Take the next digit (5) from the dividend and append it to 2, then repeat: 25 / 45

Moving on to step 2:
The number 45 fits into 25 0 times, leaving a remainder of 25.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
Deduct 0 from 25 and we're left with 25.
Append the next digit (5) from the dividend to 25 and continue with: 255 / 45

On to step 3:
When dividing 255 by 45, we get 5 with a remainder of 30.
Write down 5 as next digit of the result. 
Result so far: 5.0
If we take 225 away from 255, we end up with 30.
Include the next digit (8) from the dividend after 30, then repeat: 308 / 45

Going ahead to step 4:
308 divided by 45 is 6 with a remainder of 38.
Use 6 as the next digit of our solution. 
Result so far: 56.0
The remainder is 38 after subtracting 270 from 308.
Include the next digit (3) from the dividend after 38, then repeat: 383 / 45

Let's proceed to step 5:
383 divided by 45 is 8 with a remainder of 23.
The number 8 becomes the next digit in our result. 
Result so far: 568.0
Subtracting 360 from 383 leaves us with 23.
Since there are no more digits in the dividend, we add a zero to the remainder making it 230, and continue the process.

On to step 6:
230 divided by 45 is 5 with a remainder of 5.
Write down 5 as next digit of the result. 
Result so far: 568.5
If we subtract 225 from 230, we get 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

Going ahead to step 7:
45 goes into 50 1 times with a remainder of 5.
Record the quotient 1 as the next digit in the result. 
Result so far: 568.51
Subtract 45 from 50 to get 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

Advancing to step 8:
50 divided by 45 is 1 with a remainder of 5.
Put 1 as the next digit of the answer. 
Result so far: 568.511
If we subtract 45 from 50, we get 5.
Since there are no more digits in the dividend, we add a zero to the remainder making it 50, and continue the process.

Our division results in 568.511 with a remaining 0.0005.