1585 ÷ 9 
We want to divide 1585 by 9.

Let's proceed to step 1:
If we divide 1 by 9, we get 0 and a remainder of 1.
The number 0 becomes the next digit in our result. 
Result so far: 0.0
The remainder is 1 after subtracting 0 from 1.
Take the next digit (5) from the dividend and append it to 1, then repeat: 15 / 9

Step 2:
When dividing 15 by 9, we get 1 with a remainder of 6.
Put 1 as the next digit of the answer. 
Result so far: 1.0
Subtract 9 from 15 to get 6.
Append the next digit (8) from the dividend to 6 and continue with: 68 / 9

Step 3:
68 divided by 9 is 7 with a remainder of 5.
The next digit of our result is 7. 
Result so far: 17.0
Deduct 63 from 68 and we're left with 5.
Include the next digit (5) from the dividend after 5, then repeat: 55 / 9

Advancing to step 4:
The number 9 fits into 55 6 times, leaving a remainder of 1.
Record the quotient 6 as the next digit in the result. 
Result so far: 176.0
If we subtract 54 from 55, we 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.

Let's proceed to step 5:
9 goes into 10 1 times with a remainder of 1.
The number 1 becomes the next digit in our result. 
Result so far: 176.1
Subtract 9 from 10 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.

Let's proceed to step 6:
When dividing 10 by 9, we get 1 with a remainder of 1.
The next digit of our result is 1. 
Result so far: 176.11
The remainder is 1 after subtracting 9 from 10.
Since there are no more digits in the dividend, we add a zero to the remainder making it 10, and continue the process.

The quotient of the division is 176.11, and the remainder is 0.001.