None. And any attempts to say other wise are defeated in higher math. Even 0 is considered even by way of being 0 in mod(2). (similarly, 1 is odd by being 1 in mod(2) )
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Kind of. The only Prime number that is even is 2.
They have only one thing in common. By adding 1 to either an even number or an odd number makes them opposite. (even becomes odd and odd becomes even).
None
No. The property of odd and even is only appropriate for integers.
The sum of two odd numbers never makes an odd number. Only the sum of an odd number and an even number makes an odd number.