340561 ÷ 9 
Let's see how many times 9 fits into 340561.

Going ahead to step 1:
If we divide 3 by 9, we get 0 and a remainder of 3.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
Subtracting 0 from 3 leaves us with 3.
Include the next digit (4) from the dividend after 3, then repeat: 34 / 9

Let's proceed to step 2:
9 can be fit into 34 3 times, resulting in a remainder of 7.
The next digit of our result is 3. 
Result so far: 3.0
If we take 27 away from 34, we end up with 7.
Take the next digit (0) from the dividend and append it to 7, then repeat: 70 / 9

Moving on to step 3:
9 goes into 70 7 times with a remainder of 7.
Record the quotient 7 as the next digit in the result. 
Result so far: 37.0
Subtracting 63 from 70 leaves us with 7.
Fetch the next digit (5) from the dividend, attach it to 7 and continue: 75 / 9

On to step 4:
The number 9 fits into 75 8 times, leaving a remainder of 3.
Put 8 as the next digit of the answer. 
Result so far: 378.0
If we take 72 away from 75, we end up with 3.
Take the next digit (6) from the dividend and append it to 3, then repeat: 36 / 9

Moving on to step 5:
9 goes into 36 4 times with a remainder of 0.
Use 4 as the next digit of our solution. 
Result so far: 3784.0
If we subtract 36 from 36, we get 0.
Take the next digit (1) from the dividend and append it to 0, then repeat: 1 / 9

Let's proceed to step 6:
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: 37840.0
Deduct 0 from 1 and we're left with 1.
Since there are no more digits in the dividend, we add a zero to the remainder making it 10, and continue the process.

Moving on to step 7:
10 divided by 9 is 1 with a remainder of 1.
The next digit of our result is 1. 
Result so far: 37840.1
Deduct 9 from 10 and we're left with 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 8:
9 can be fit into 10 1 times, resulting in a remainder of 1.
Write down 1 as next digit of the result. 
Result so far: 37840.11
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.

The quotient of the division is 37840.11, and the remainder is 0.001.