19259 ÷ 4 = 4814 R3
Alright, let's work through the division of 19259 by 4 step by step.

Step 1:
4 goes into 1 0 times with a remainder of 1.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 1 to get 1.
Bring next digit (9) of the dividend behind the 1 and repeat the process: 19 / 4

Step 2:
4 goes into 19 4 times with a remainder of 3.
Write down 4 as next digit of of the result. 
Result so far: 4
Subtract 16 from 19 to get 3.
Bring next digit (2) of the dividend behind the 3 and repeat the process: 32 / 4

Step 3:
4 goes into 32 8 times with a remainder of 0.
Write down 8 as next digit of of the result. 
Result so far: 48
Subtract 32 from 32 to get 0.
Bring next digit (5) of the dividend behind the 0 and repeat the process: 5 / 4

Step 4:
4 goes into 5 1 times with a remainder of 1.
Write down 1 as next digit of of the result. 
Result so far: 481
Subtract 4 from 5 to get 1.
Bring next digit (9) of the dividend behind the 1 and repeat the process: 19 / 4

Step 5:
4 goes into 19 4 times with a remainder of 3.
Write down 4 as next digit of of the result. 
Result so far: 4814
Subtract 16 from 19 to get 3.

The final result is 4814 with a remainder of 3.