Any integer can be divided by any non-zero integer, and the result is a rational number.
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They are not the same!The set of integers is closed under multiplication but not under division.Multiplication is commutative, division is not.Multiplication is associative, division is not.
The rules are not the same.Multiplication is commutative whereas division is not.Multiplication is associative whereas division is not.
Those are the rules of multiplication (and division).
I am not at all sure that there are any rules that apply to integers in isolation. Any rules that exist are in the context of binary operations like addition or multiplication of integers.
I am not sure there are any fundamental operations of integers. The fundamental operations of arithmetic are addition, subtraction, multiplication and division. However, the set of integers is not closed with respect to division: that is, the division of one integer by another does not necessarily result in an integer.