Why does 1366x768 resolution exist?

TL;DR: Numerous reasons:

1. 1024768 was already the XGA standard, and increasing WXGA to 1366768 was the easiest way to do this and retain the height of 768. If the height changed, other (small) things would have had to be changed.

2. The bizarre aspect ratio, 683:384, reduces to 16.008:9, which is very close to the common aspect ratio of 16:9. 1365.33768 would give 16:9 and a fractional pixel.

3. The resolution of 1360768 is uncommon, but video data can be stored entirely in an 8MiB memory chip. Adding the extra 6 lines requires a slightly more expensive 16MiB chip. Not sure why they didn't go with 1360*768.