76584 ÷ 6 = 12764 R0
Alright, let's work through the division of 76584 by 6 step by step.

Step 1:
6 goes into 7 1 times with a remainder of 1.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 6 from 7 to get 1.
Bring next digit (6) of the dividend behind the 1 and repeat the process: 16 / 6

Step 2:
6 goes into 16 2 times with a remainder of 4.
Write down 2 as next digit of of the result. 
Result so far: 12
Subtract 12 from 16 to get 4.
Bring next digit (5) of the dividend behind the 4 and repeat the process: 45 / 6

Step 3:
6 goes into 45 7 times with a remainder of 3.
Write down 7 as next digit of of the result. 
Result so far: 127
Subtract 42 from 45 to get 3.
Bring next digit (8) of the dividend behind the 3 and repeat the process: 38 / 6

Step 4:
6 goes into 38 6 times with a remainder of 2.
Write down 6 as next digit of of the result. 
Result so far: 1276
Subtract 36 from 38 to get 2.
Bring next digit (4) of the dividend behind the 2 and repeat the process: 24 / 6

Step 5:
6 goes into 24 4 times with a remainder of 0.
Write down 4 as next digit of of the result. 
Result so far: 12764
Subtract 24 from 24 to get 0.

The final result is 12764 with a remainder of 0.