Should I set the DPI on web images to 72 DPI?

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Set it to whatever you want. Make it 1 or 20,000. It makes no difference in image quality or file size. Unless your final output is print, dpi/ppi is irrelevant. Your monitor only understands pixels. If you don't re-sample, a 72 dpi images is going to be the same as a 300 dpi image. Un-check re-sample in PS and change the dpi and you'll see the pixels don't change.

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