Even or odd is a property that only integers (the whole numbers, including positives, negatives, and zero) have. We do not call other numbers (fractions, Irrational Numbers, imaginary numbers, and so on) either even or odd.
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Any number that is not an integer is neither even nor odd. For example, 8.5, pi, the square root of 2 are all neither even nor odd. Every integer (including 0) is either even or odd. 0 is even since it is divisible by 2: 0 / 2 = 0
Odd. Even + Even = Even Odd + Odd = Even Odd + Even = Even + Odd = Odd
odd * odd = odd answer even * even = even answer odd * even = even answer
Yes. Even + Even = Even, Odd + Odd = Even and Even + Odd or Odd + Even = Odd
I do not understand the significance of the word "riddle" in the question, but related to math, non-integers are neither odd nor even. For example, 1.5 and pi are neither even nor odd. 0 is an even number. See "Parity of Zero" in the Related Links for a discussion of why many people mistakenly believe 0 is not even.