There is no reason to give, because that's not a true statement.
Examples:
There is no integer between 4 and 5, or between 27 and 28, or between 792 and 793.
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Yes the sum of two integers will always be an integer.
The statement is false.
It is two times the magnitude of the integer.
The statement is: "Their sum is always an integer." Whether the sum is negative or positive depends on the two original integers.
Yes, the sum of two negative integers is always negative.