77557 ÷ 7 = 11079 R4
Alright, let's work through the division of 77557 by 7 step by step.

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

Step 2:
7 goes into 7 1 times with a remainder of 0.
Write down 1 as next digit of of the result. 
Result so far: 11
Subtract 7 from 7 to get 0.
Bring next digit (5) of the dividend behind the 0 and repeat the process: 5 / 7

Step 3:
7 goes into 5 0 times with a remainder of 5.
Write down 0 as next digit of of the result. 
Result so far: 110
Subtract 0 from 5 to get 5.
Bring next digit (5) of the dividend behind the 5 and repeat the process: 55 / 7

Step 4:
7 goes into 55 7 times with a remainder of 6.
Write down 7 as next digit of of the result. 
Result so far: 1107
Subtract 49 from 55 to get 6.
Bring next digit (7) of the dividend behind the 6 and repeat the process: 67 / 7

Step 5:
7 goes into 67 9 times with a remainder of 4.
Write down 9 as next digit of of the result. 
Result so far: 11079
Subtract 63 from 67 to get 4.

The final result is 11079 with a remainder of 4.