We look at the division of 8354430 by 603 
We want to figure out the number of times 8354430 can be divided by 603.

Advancing to step 1:
The number 603 fits into 8 0 times, leaving a remainder of 8.
Write down 0 as next digit of the result. 
Result so far: 0.0
Deduct 0 from 8 and we're left with 8.
Append the next digit (3) from the dividend to 8 and continue with: 83 / 603

On to step 2:
If we divide 83 by 603, we get 0 and a remainder of 83.
Write down 0 as next digit of the result. 
Result so far: 0.0
Subtract 0 from 83 to get 83.
Take the next digit (5) from the dividend and append it to 83, then repeat: 835 / 603

On to step 3:
If we divide 835 by 603, we get 1 and a remainder of 232.
Put 1 as the next digit of the answer. 
Result so far: 1.0
If we subtract 603 from 835, we get 232.
Bring next digit (4) of the dividend behind the 232 and repeat the process: 2324 / 603

Advancing to step 4:
If we divide 2324 by 603, we get 3 and a remainder of 515.
Write down 3 as next digit of the result. 
Result so far: 13.0
Deduct 1809 from 2324 and we're left with 515.
Append the next digit (4) from the dividend to 515 and continue with: 5154 / 603

On to step 5:
If we divide 5154 by 603, we get 8 and a remainder of 330.
The number 8 becomes the next digit in our result. 
Result so far: 138.0
If we subtract 4824 from 5154, we get 330.
Include the next digit (3) from the dividend after 330, then repeat: 3303 / 603

Advancing to step 6:
When dividing 3303 by 603, we get 5 with a remainder of 288.
Write down 5 as next digit of the result. 
Result so far: 1385.0
If we subtract 3015 from 3303, we get 288.
Bring next digit (0) of the dividend behind the 288 and repeat the process: 2880 / 603

Let's proceed to step 7:
603 goes into 2880 4 times with a remainder of 468.
Record the quotient 4 as the next digit in the result. 
Result so far: 13854.0
If we subtract 2412 from 2880, we get 468.
Since there are no more digits in the dividend, we add a zero to the remainder making it 4680, and continue the process.

Let's proceed to step 8:
When dividing 4680 by 603, we get 7 with a remainder of 459.
Write down 7 as next digit of the result. 
Result so far: 13854.7
Subtracting 4221 from 4680 leaves us with 459.
Since there are no more digits in the dividend, we add a zero to the remainder making it 4590, and continue the process.

Moving on to step 9:
When dividing 4590 by 603, we get 7 with a remainder of 369.
Write down 7 as next digit of the result. 
Result so far: 13854.77
Subtracting 4221 from 4590 leaves us with 369.
Since there are no more digits in the dividend, we add a zero to the remainder making it 3690, and continue the process.

The quotient of the division is 13854.77, and the remainder is 0.369.