Downloading anything for yourself on someone else's Xbox is always a bad idea. Theres some crap with DLC that makes it so it associates it with your Gamer tag AND the console, which then makes it so you can still play it on other consoles, however you need to be connected to Xbox Live and signed into the profile used to purchase the content.
Its a process a lot of people complain to Xbox Support about, believe me. (I used to work for Xbox 360 Support...it was a nightmare.)
Also just to make it clear, it associates it with your CONSOLE not the HARD DRIVE.
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I did have something here about needing more Dave Grohl...but...well we got more Dave Grohl...