An overheated CPU is the most likely cause of a computer that has been randomly shutting down without warning. A CPU generates heat when it performs calculations, and if the heat is not dissipated properly, it can damage the CPU or cause it to malfunction. A computer that shuts down randomly is a common symptom of an overheated CPU, as the system tries to protect itself from further damage by shutting down or restarting.

The other options are not as likely to cause this issue. A corrupted OS may cause the computer to crash, display errors, or fail to boot, but it does not affect the CPU temperature or the system fan. Incompatible memory modules may cause the computer to beep, display a blue screen, or fail to recognize the installed RAM, but they do not cause the CPU to overheat or the system to shut down randomly. Low system resources may cause the computer to slow down, freeze, or display a low memory warning, but they do not affect the CPU temperature or the system fan.

To troubleshoot and resolve an overheated CPU issue, the user should check and clean the air vents, provide proper ventilation, apply thermal paste, replace the fan or the heat sink, or upgrade the CPU if necessary. The user should also update the BIOS and the device drivers, adjust the power settings, and monitor the system temperature using diagnostic tools.