How many bits would need to show on a binary odometer if you wanted to have the ability to count up to 500?

The binary system count 0 and before shifting left.
Therefore repeated divisions of 500 by 2 will yield the bits as reminders.
 
500/2 = 250 R 0
250/2 =  125 R 0
125/2  =   62 R  1
  62/2 =    31  R 0
  31/2  =    15  R 1
  15/2  =     7  R 1
   7/2  =      3 R 1
   3/2  =     1   R 1
   1/2  =      0  R 1

Collect the remainders to defne the bnary digits.
500 = 1 1 1 1 1 0 1 0 0
We need 9 bits, defined by the bnary number that will show on a binary odometer.

1 1 1 1 1 0 1 0 0  (9 bits on an odometer)