35437 ÷ 4 = 8859 R1
Alright, let's work through the division of 35437 by 4 step by step.

Step 1:
4 goes into 3 0 times with a remainder of 3.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 3 to get 3.
Bring next digit (5) of the dividend behind the 3 and repeat the process: 35 / 4

Step 2:
4 goes into 35 8 times with a remainder of 3.
Write down 8 as next digit of of the result. 
Result so far: 8
Subtract 32 from 35 to get 3.
Bring next digit (4) of the dividend behind the 3 and repeat the process: 34 / 4

Step 3:
4 goes into 34 8 times with a remainder of 2.
Write down 8 as next digit of of the result. 
Result so far: 88
Subtract 32 from 34 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: 885
Subtract 20 from 23 to get 3.
Bring next digit (7) of the dividend behind the 3 and repeat the process: 37 / 4

Step 5:
4 goes into 37 9 times with a remainder of 1.
Write down 9 as next digit of of the result. 
Result so far: 8859
Subtract 36 from 37 to get 1.

The final result is 8859 with a remainder of 1.