66577 ÷ 7 = 9511 R0
Alright, let's work through the division of 66577 by 7 step by step.

Step 1:
7 goes into 6 0 times with a remainder of 6.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 6 to get 6.
Bring next digit (6) of the dividend behind the 6 and repeat the process: 66 / 7

Step 2:
7 goes into 66 9 times with a remainder of 3.
Write down 9 as next digit of of the result. 
Result so far: 9
Subtract 63 from 66 to get 3.
Bring next digit (5) of the dividend behind the 3 and repeat the process: 35 / 7

Step 3:
7 goes into 35 5 times with a remainder of 0.
Write down 5 as next digit of of the result. 
Result so far: 95
Subtract 35 from 35 to get 0.
Bring next digit (7) of the dividend behind the 0 and repeat the process: 7 / 7

Step 4:
7 goes into 7 1 times with a remainder of 0.
Write down 1 as next digit of of the result. 
Result so far: 951
Subtract 7 from 7 to get 0.
Bring next digit (7) of the dividend behind the 0 and repeat the process: 7 / 7

Step 5:
7 goes into 7 1 times with a remainder of 0.
Write down 1 as next digit of of the result. 
Result so far: 9511
Subtract 7 from 7 to get 0.

The final result is 9511 with a remainder of 0.