3817 ÷ 6 = 636 R1
Alright, let's work through the division of 3817 by 6 step by step.

Step 1:
6 goes into 3 0 times with a remainder of 3.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 3 to get 3.
Bring next digit (8) of the dividend behind the 3 and repeat the process: 38 / 6

Step 2:
6 goes into 38 6 times with a remainder of 2.
Write down 6 as next digit of of the result. 
Result so far: 6
Subtract 36 from 38 to get 2.
Bring next digit (1) of the dividend behind the 2 and repeat the process: 21 / 6

Step 3:
6 goes into 21 3 times with a remainder of 3.
Write down 3 as next digit of of the result. 
Result so far: 63
Subtract 18 from 21 to get 3.
Bring next digit (7) of the dividend behind the 3 and repeat the process: 37 / 6

Step 4:
6 goes into 37 6 times with a remainder of 1.
Write down 6 as next digit of of the result. 
Result so far: 636
Subtract 36 from 37 to get 1.

The final result is 636 with a remainder of 1.