240.8
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360/800 and 36000/80000
360/810, 3636/8181 are two.
Because it has so many factors. That means you can easily break 360 down into equal pieces without the need to mess with fractions. Pick any number from the following list. 360 can be divided up into that many pieces without any fractions of a degree: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, or 180 .
It is believed that the ancient Babylonians divided the circle into 360 degrees. 360 has the advantage that it is divisible by many of the numbers used for fractions used in everyday life such as 2, 3, 4, 5, 6, 8, 9, 10, 12 and 60. The fact the earth did a complete circle around the sun in nearly 360 days might have helped.
Basically because it was decided that way, there are plenty of other numbers that could have been used. With that said, 360 is a good choice, it divides evenly into lot of fractions and it has a resolution that is just about what's need for many applications.