We're dividing 421106 by 3 
We want to figure out the number of times 421106 can be divided by 3.

Moving on to step 1:
When dividing 4 by 3, we get 1 with a remainder of 1.
Record the quotient 1 as the next digit in the result. 
Result so far: 1.0
Subtracting 3 from 4 leaves us with 1.
Fetch the next digit (2) from the dividend, attach it to 1 and continue: 12 / 3

Step 2:
If we divide 12 by 3, we get 4 and a remainder of 0.
The number 4 becomes the next digit in our result. 
Result so far: 14.0
The remainder is 0 after subtracting 12 from 12.
Bring next digit (1) of the dividend behind the 0 and repeat the process: 1 / 3

Going ahead to step 3:
When dividing 1 by 3, we get 0 with a remainder of 1.
Write down 0 as next digit of the result. 
Result so far: 140.0
Subtracting 0 from 1 leaves us with 1.
Include the next digit (1) from the dividend after 1, then repeat: 11 / 3

Let's proceed to step 4:
11 divided by 3 is 3 with a remainder of 2.
Write down 3 as next digit of the result. 
Result so far: 1403.0
Deduct 9 from 11 and we're left with 2.
Append the next digit (0) from the dividend to 2 and continue with: 20 / 3

On to step 5:
When dividing 20 by 3, we get 6 with a remainder of 2.
Write down 6 as next digit of the result. 
Result so far: 14036.0
Deduct 18 from 20 and we're left with 2.
Grab the next digit (6) from the dividend, add it to 2, then carry on: 26 / 3

On to step 6:
3 can be fit into 26 8 times, resulting in a remainder of 2.
Use 8 as the next digit of our solution. 
Result so far: 140368.0
Subtract 24 from 26 to get 2.
Since there are no more digits in the dividend, we add a zero to the remainder making it 20, and continue the process.

Let's proceed to step 7:
20 divided by 3 is 6 with a remainder of 2.
Record the quotient 6 as the next digit in the result. 
Result so far: 140368.6
The remainder is 2 after subtracting 18 from 20.
Since there are no more digits in the dividend, we add a zero to the remainder making it 20, and continue the process.

Step 8:
If we divide 20 by 3, we get 6 and a remainder of 2.
Use 6 as the next digit of our solution. 
Result so far: 140368.66
Subtracting 18 from 20 leaves us 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.

After the division, we end up with 140368.66 and a remainder of 0.002.