8583 ÷ 4 = 2145 R3
Alright, let's work through the division of 8583 by 4 step by step.

Step 1:
4 goes into 8 2 times with a remainder of 0.
Write down 2 as next digit of of the result. 
Result so far: 2
Subtract 8 from 8 to get 0.
Bring next digit (5) of the dividend behind the 0 and repeat the process: 5 / 4

Step 2:
4 goes into 5 1 times with a remainder of 1.
Write down 1 as next digit of of the result. 
Result so far: 21
Subtract 4 from 5 to get 1.
Bring next digit (8) of the dividend behind the 1 and repeat the process: 18 / 4

Step 3:
4 goes into 18 4 times with a remainder of 2.
Write down 4 as next digit of of the result. 
Result so far: 214
Subtract 16 from 18 to get 2.
Bring next digit (3) of the dividend behind the 2 and repeat the process: 23 / 4

Step 4:
4 goes into 23 5 times with a remainder of 3.
Write down 5 as next digit of of the result. 
Result so far: 2145
Subtract 20 from 23 to get 3.

The final result is 2145 with a remainder of 3.