15538 ÷ 6 = 2589 R4
Alright, let's work through the division of 15538 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 (5) of the dividend behind the 1 and repeat the process: 15 / 6

Step 2:
6 goes into 15 2 times with a remainder of 3.
Write down 2 as next digit of of the result. 
Result so far: 2
Subtract 12 from 15 to get 3.
Bring next digit (5) of the dividend behind the 3 and repeat the process: 35 / 6

Step 3:
6 goes into 35 5 times with a remainder of 5.
Write down 5 as next digit of of the result. 
Result so far: 25
Subtract 30 from 35 to get 5.
Bring next digit (3) of the dividend behind the 5 and repeat the process: 53 / 6

Step 4:
6 goes into 53 8 times with a remainder of 5.
Write down 8 as next digit of of the result. 
Result so far: 258
Subtract 48 from 53 to get 5.
Bring next digit (8) of the dividend behind the 5 and repeat the process: 58 / 6

Step 5:
6 goes into 58 9 times with a remainder of 4.
Write down 9 as next digit of of the result. 
Result so far: 2589
Subtract 54 from 58 to get 4.

The final result is 2589 with a remainder of 4.