41037 ÷ 9 = 4559 R6
Alright, let's work through the division of 41037 by 9 step by step.

Step 1:
9 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 (1) of the dividend behind the 4 and repeat the process: 41 / 9

Step 2:
9 goes into 41 4 times with a remainder of 5.
Write down 4 as next digit of of the result. 
Result so far: 4
Subtract 36 from 41 to get 5.
Bring next digit (0) of the dividend behind the 5 and repeat the process: 50 / 9

Step 3:
9 goes into 50 5 times with a remainder of 5.
Write down 5 as next digit of of the result. 
Result so far: 45
Subtract 45 from 50 to get 5.
Bring next digit (3) of the dividend behind the 5 and repeat the process: 53 / 9

Step 4:
9 goes into 53 5 times with a remainder of 8.
Write down 5 as next digit of of the result. 
Result so far: 455
Subtract 45 from 53 to get 8.
Bring next digit (7) of the dividend behind the 8 and repeat the process: 87 / 9

Step 5:
9 goes into 87 9 times with a remainder of 6.
Write down 9 as next digit of of the result. 
Result so far: 4559
Subtract 81 from 87 to get 6.

The final result is 4559 with a remainder of 6.