No, it is not.
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If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.
No.
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No, but they are closed for multiplication.
If you interpret "whole numbers" as "integers", then yes. If you interpret "whole numbers" as "non-negative integers", then no.