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Basics
The E76 error is caused by a faulty ethernet network chip. The ethernet chip is what regulates the Xbox 360's ethernet capabilities, which allow users to connect to Xbox Live and play online. The chip can be rendered faulty by an overload of electricity brought on by power surges, particularly during lightning storms.
Indication
The Xbox 360's power lights, which are normally green, will go out and be replaced by a single flashing red light in the bottom right corner. This indicates numerous errors, including the E76 ethernet chip error. "Error 76" will flash when you attempt to boot up the console.
Troubleshooting
It is possible that the error will work itself out, and your ethernet chip may not be faulty. The Xbox 360 may have misread the chip's status, resulting in the error code. Power your unit on and off until it boots up normally. Power off the console and disconnect all cords, memory units and the hard drive from your console if it still does not work. Reconnect everything and try powering it on again. If it still does not work, it may need servicing.
Repair
The E76 error is fixed by replacing the dead ethernet controller chip. You can attempt to do this yourself by referring to an Xbox 360 repair guide. This will void your warranty, however. The procedure involves disassembling the unit, prying off the old ethernet chip and swapping in a new one. You should contact Microsoft for servicing if you do not want to void your warranty or do not feel comfortable repairing the unit yourself. Microsoft will repair the console for free if it is still under warranty and for a fee if it is not.