If I Download a Game Onto My Xbox 360 Hard Drive with a Good Disk, Will It Let Me Play It with a Scratched One?

It's kinda self explanatory, but i'll add more detail:

I have a copy of Call of Duty: Modern Warfare 2. It had a ring burned into it. I'm wondering… if I go to the local video store and rent a copy of the same game, if I download the game to my Xbox's hard drive, will the Xbox allow me to play the game with a scratched version of the disk? I assumed that since it's on the hard drive it kind of just needs the disk to "run the program" so to speak in computer terms. Kind of like a computer game where it's on the drive, but you need the disk to start it and keep it running? I hope that's what it's like. If anyone can let me know if this will work, it would make my day Thank you in advance
– Dave

Suggestion:

It really depends on how damaged your disc is. Just send it to cdfixers.com. They fix it for a few bucks guaranteed.

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