40003 ÷ 6 = 6667 R1
Alright, let's work through the division of 40003 by 6 step by step.

Step 1:
6 goes into 4 0 times with a remainder of 4.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 4 to get 4.
Bring next digit (0) of the dividend behind the 4 and repeat the process: 40 / 6

Step 2:
6 goes into 40 6 times with a remainder of 4.
Write down 6 as next digit of of the result. 
Result so far: 6
Subtract 36 from 40 to get 4.
Bring next digit (0) of the dividend behind the 4 and repeat the process: 40 / 6

Step 3:
6 goes into 40 6 times with a remainder of 4.
Write down 6 as next digit of of the result. 
Result so far: 66
Subtract 36 from 40 to get 4.
Bring next digit (0) of the dividend behind the 4 and repeat the process: 40 / 6

Step 4:
6 goes into 40 6 times with a remainder of 4.
Write down 6 as next digit of of the result. 
Result so far: 666
Subtract 36 from 40 to get 4.
Bring next digit (3) of the dividend behind the 4 and repeat the process: 43 / 6

Step 5:
6 goes into 43 7 times with a remainder of 1.
Write down 7 as next digit of of the result. 
Result so far: 6667
Subtract 42 from 43 to get 1.

The final result is 6667 with a remainder of 1.