Yes, they ARE.
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That is how whole numbers are defined.
Integers are defined as whole numbers.
Yes. That means that the product of two whole numbers is defined, and that it is again a whole number.
The set is well defined. Whether or not a given integer belongs to the set of prime numbers is clearly defined even if, for extremely large numbers, it may prove impossible to determine the status of that number.
Prime numbers have only 2 factors and their set is not well defined because they do not follow an orderly mathematical pattern.