Radio turns on when car is off and keys are out

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Radio/clock display turns on when car is not on and keys are out of ignition (and if I accidentally leave the radio on when I turn the car off, the radio turns on and off too), causing it to slowly drain the battery. Had to take fuse out of it (which is also the fuse for the lights to AC, etc.) When car is on and running, the radio is fine--it does not cut off or mess up in any way.