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Will I be able to use Xbox Live® if I don't have a hard drive?
You won't need a hard drive to enjoy all the benefits of Xbox Live Silver and Gold with the Core System, but you will at least need a Memory Unit (MU) in order to save your Gamer Profile. If you plan on downloading a lot of content (vehicles, maps, game trailers, demos, etc) you'll want to get a hard drive.
We were discussing this question on another board and I'm not quite sure that the answer is so cut and dry. " you will at least need a Memory Unit (MU) in order to save your Gamer Profile" we know the "gamer Profile" can't be saved without a HDD or MU card but will it run in memory if it's recovered everytime before starting live. It almost resembles how a cookie works in most web browsers but in this case when the system shuts down the "cookie" is erased.