81001 ÷ 9 = 9000 R1
Alright, let's work through the division of 81001 by 9 step by step.

Step 1:
9 goes into 8 0 times with a remainder of 8.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 8 to get 8.
Bring next digit (1) of the dividend behind the 8 and repeat the process: 81 / 9

Step 2:
9 goes into 81 9 times with a remainder of 0.
Write down 9 as next digit of of the result. 
Result so far: 9
Subtract 81 from 81 to get 0.
Bring next digit (0) of the dividend behind the 0 and repeat the process: 0 / 9

Step 3:
9 goes into 0 0 times with a remainder of 0.
Write down 0 as next digit of of the result. 
Result so far: 90
Subtract 0 from 0 to get 0.
Bring next digit (0) of the dividend behind the 0 and repeat the process: 0 / 9

Step 4:
9 goes into 0 0 times with a remainder of 0.
Write down 0 as next digit of of the result. 
Result so far: 900
Subtract 0 from 0 to get 0.
Bring next digit (1) of the dividend behind the 0 and repeat the process: 1 / 9

Step 5:
9 goes into 1 0 times with a remainder of 1.
Write down 0 as next digit of of the result. 
Result so far: 9000
Subtract 0 from 1 to get 1.

The final result is 9000 with a remainder of 1.