Is there a quality, file-size, or other benefit to JPEG sizes being multiples of 8px or 16px?

8 pixels is the cutoff. The reason is because JPEG images are simply an array of 8x8 DCT blocks; if the image resolution isn't mod8 in both directions, the encoder has to pad the sides up to the next mod8 resolution. This in practice is not very expensive bit-wise; what's much worse are the cases when an image has sharp black lines (such as a letterboxed image) that don't lie on block boundaries.

This is especially problematic in video encoding. The reason for this being a problem is that the frequency transform of a sharp line is a Gaussian distribution of coefficients--resulting in an enormous number of bits to code.

A JPG with sizes being multiplies of 8 can also be rotated/flipped with no quality loss. For example gthumb can do this on Linux.

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