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I've literally just had EXACTLY the same problem on my brother's old PC.
It was quite easily cured by completely powering off (including removing the power supply from the wall), removing the memory modules and cleaning the contacts with a soft cloth lightly soaked in ethanol. I then cleaned out the memory slots of dust and dirt with a fine, dry artist's paintbrush.
About five minutes later, ensuring the contacts were dry, I replaced them and voila, a working PC again!