Do 1440p 60hz monitor run 1080p 144fps or 240fps

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No, not natively you could probably overclock it past 60hz at 1080p but I doubt you would get anywhere near 144hz. And 2560x1440p native monitors really don't look that great at 1920x1080p, it's not the same as a native 1080p monitor it looks more blurry.
No, not natively you could probably overclock it past 60hz at 1080p but I doubt you would get anywhere near 144hz. And 2560x1440p native monitors really don't look that great at 1920x1080p, it's not the same as a native 1080p monitor it looks more blurry.
 
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