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If I unplug my Xbox 360 and plug it back into the same place, what happens to my GTA 5 progress if it's saved? Will my game load to where it left off at?
The Xbox 360 hard drive will keep your data even if you switch off or unplug the Xbox. That's what it's for.
As long as you save your game before unplugging the Xbox, you'll be able to continue where you left off.