Let $a$ and $b$ two positive integers such that $b≠0$.
How one can makes a difference between integers and rational numbers by using the form $q=a/b$, i.e., cases of the form:
$q$ is an integer if and only if conditions on $a$ and $b$.
$q$ is a rational number and not an integer if and only if conditions on $a$ and $b$.
Every integer $n\in\Bbb Z$ is also a rational number, since one way of writing it as a fraction is $\frac{n}{1}$.
If you want do distinguish between rational numbers that are and aren't integers you could have,
$$q=\frac{a}{b}\text{ is an integer }\iff b\mid a$$