513 ÷ 9 = 57 R0
Alright, let's work through the division of 513 by 9 step by step.

Step 1:
9 goes into 5 0 times with a remainder of 5.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 5 to get 5.
Bring next digit (1) of the dividend behind the 5 and repeat the process: 51 / 9

Step 2:
9 goes into 51 5 times with a remainder of 6.
Write down 5 as next digit of of the result. 
Result so far: 5
Subtract 45 from 51 to get 6.
Bring next digit (3) of the dividend behind the 6 and repeat the process: 63 / 9

Step 3:
9 goes into 63 7 times with a remainder of 0.
Write down 7 as next digit of of the result. 
Result so far: 57
Subtract 63 from 63 to get 0.

The final result is 57 with a remainder of 0.