Yes, because the concept of odd and even is defined only for integers.
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Every integer is a rational number.
No, not every negative number is an integer. For example, -11/2 is not an integer. However, -1, -2, -3, and so on, are negative integers. Perhaps that is what you meant to ask. The negative of every positive integer is a negative integer.
Every counting number IS an integer.
No, but every whole is a integer. integers are - and +. whole number are only +.
Every integer is all three.