4377 ÷ 2 = 2188 R1
Alright, let's work through the division of 4377 by 2 step by step.

Step 1:
2 goes into 4 2 times with a remainder of 0.
Write down 2 as next digit of of the result. 
Result so far: 2
Subtract 4 from 4 to get 0.
Bring next digit (3) of the dividend behind the 0 and repeat the process: 3 / 2

Step 2:
2 goes into 3 1 times with a remainder of 1.
Write down 1 as next digit of of the result. 
Result so far: 21
Subtract 2 from 3 to get 1.
Bring next digit (7) of the dividend behind the 1 and repeat the process: 17 / 2

Step 3:
2 goes into 17 8 times with a remainder of 1.
Write down 8 as next digit of of the result. 
Result so far: 218
Subtract 16 from 17 to get 1.
Bring next digit (7) of the dividend behind the 1 and repeat the process: 17 / 2

Step 4:
2 goes into 17 8 times with a remainder of 1.
Write down 8 as next digit of of the result. 
Result so far: 2188
Subtract 16 from 17 to get 1.

The final result is 2188 with a remainder of 1.