It is not possible to have the product of an integer. "product" is a binary operation and that means that it is an operation that combines two numbers to make the product - a third number. So you need two numbers as input, not just one.
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Yes. The product of a negative integer and a positive integer is a negative integer.
Always.
positive
perfect square
Yes, by definition, the sum of two integers is always an integer. Likewise, the product and difference of two integers is always an integer.