13638 ÷ 6 = 2273 R0
Alright, let's work through the division of 13638 by 6 step by step.

Step 1:
6 goes into 1 0 times with a remainder of 1.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 1 to get 1.
Bring next digit (3) of the dividend behind the 1 and repeat the process: 13 / 6

Step 2:
6 goes into 13 2 times with a remainder of 1.
Write down 2 as next digit of of the result. 
Result so far: 2
Subtract 12 from 13 to get 1.
Bring next digit (6) of the dividend behind the 1 and repeat the process: 16 / 6

Step 3:
6 goes into 16 2 times with a remainder of 4.
Write down 2 as next digit of of the result. 
Result so far: 22
Subtract 12 from 16 to get 4.
Bring next digit (3) of the dividend behind the 4 and repeat the process: 43 / 6

Step 4:
6 goes into 43 7 times with a remainder of 1.
Write down 7 as next digit of of the result. 
Result so far: 227
Subtract 42 from 43 to get 1.
Bring next digit (8) of the dividend behind the 1 and repeat the process: 18 / 6

Step 5:
6 goes into 18 3 times with a remainder of 0.
Write down 3 as next digit of of the result. 
Result so far: 2273
Subtract 18 from 18 to get 0.

The final result is 2273 with a remainder of 0.