10219 ÷ 3 = 3406 R1
Alright, let's work through the division of 10219 by 3 step by step.

Step 1:
3 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 (0) of the dividend behind the 1 and repeat the process: 10 / 3

Step 2:
3 goes into 10 3 times with a remainder of 1.
Write down 3 as next digit of of the result. 
Result so far: 3
Subtract 9 from 10 to get 1.
Bring next digit (2) of the dividend behind the 1 and repeat the process: 12 / 3

Step 3:
3 goes into 12 4 times with a remainder of 0.
Write down 4 as next digit of of the result. 
Result so far: 34
Subtract 12 from 12 to get 0.
Bring next digit (1) of the dividend behind the 0 and repeat the process: 1 / 3

Step 4:
3 goes into 1 0 times with a remainder of 1.
Write down 0 as next digit of of the result. 
Result so far: 340
Subtract 0 from 1 to get 1.
Bring next digit (9) of the dividend behind the 1 and repeat the process: 19 / 3

Step 5:
3 goes into 19 6 times with a remainder of 1.
Write down 6 as next digit of of the result. 
Result so far: 3406
Subtract 18 from 19 to get 1.

The final result is 3406 with a remainder of 1.