No. A simple definition of "integer" is "whole number", that is, a number without a fractional part.
The rational number doesn't have to be a whole number, but if it is, then it's an integer.
A rational number that is a whole number is also an integer. Therefore, it is not possible for a number to be a rational number that is not an integer but still be considered a whole number. The definitions of these number sets overlap in a way that prevents such a scenario.
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