25586 ÷ 9 = 2842 R8
Alright, let's work through the division of 25586 by 9 step by step.

Step 1:
9 goes into 2 0 times with a remainder of 2.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 2 to get 2.
Bring next digit (5) of the dividend behind the 2 and repeat the process: 25 / 9

Step 2:
9 goes into 25 2 times with a remainder of 7.
Write down 2 as next digit of of the result. 
Result so far: 2
Subtract 18 from 25 to get 7.
Bring next digit (5) of the dividend behind the 7 and repeat the process: 75 / 9

Step 3:
9 goes into 75 8 times with a remainder of 3.
Write down 8 as next digit of of the result. 
Result so far: 28
Subtract 72 from 75 to get 3.
Bring next digit (8) of the dividend behind the 3 and repeat the process: 38 / 9

Step 4:
9 goes into 38 4 times with a remainder of 2.
Write down 4 as next digit of of the result. 
Result so far: 284
Subtract 36 from 38 to get 2.
Bring next digit (6) of the dividend behind the 2 and repeat the process: 26 / 9

Step 5:
9 goes into 26 2 times with a remainder of 8.
Write down 2 as next digit of of the result. 
Result so far: 2842
Subtract 18 from 26 to get 8.

The final result is 2842 with a remainder of 8.