2623 ÷ 4 = 655 R3
Alright, let's work through the division of 2623 by 4 step by step.

Step 1:
4 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 (6) of the dividend behind the 2 and repeat the process: 26 / 4

Step 2:
4 goes into 26 6 times with a remainder of 2.
Write down 6 as next digit of of the result. 
Result so far: 6
Subtract 24 from 26 to get 2.
Bring next digit (2) of the dividend behind the 2 and repeat the process: 22 / 4

Step 3:
4 goes into 22 5 times with a remainder of 2.
Write down 5 as next digit of of the result. 
Result so far: 65
Subtract 20 from 22 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: 655
Subtract 20 from 23 to get 3.

The final result is 655 with a remainder of 3.