13568 ÷ 6 = 2261 R2
Alright, let's work through the division of 13568 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 (5) of the dividend behind the 1 and repeat the process: 15 / 6

Step 3:
6 goes into 15 2 times with a remainder of 3.
Write down 2 as next digit of of the result. 
Result so far: 22
Subtract 12 from 15 to get 3.
Bring next digit (6) of the dividend behind the 3 and repeat the process: 36 / 6

Step 4:
6 goes into 36 6 times with a remainder of 0.
Write down 6 as next digit of of the result. 
Result so far: 226
Subtract 36 from 36 to get 0.
Bring next digit (8) of the dividend behind the 0 and repeat the process: 8 / 6

Step 5:
6 goes into 8 1 times with a remainder of 2.
Write down 1 as next digit of of the result. 
Result so far: 2261
Subtract 6 from 8 to get 2.

The final result is 2261 with a remainder of 2.