The integers are all the natural numbers and their negatives and zero.
So 5 is a natural number and negative 5 is its negative. That makes it an integer.
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A negative integer multiplied by a negative integer is always a positive integer product. -x * -y = xy
Yes, and also a negative integer divided by a negative integer is equal to a positive number (but not necessarily an integer).
As long as the negative integer is greater than the positive integer, a negative integer will result from addition, subtraction, multiplication, and division.