For a rational number $r \in \mathbb{Q}$, the claim is that we can write $$ r = p^{e}\cdot\frac{a}{b}, \,\,\, e,a,b \in \mathbb{Z}, b \neq 0 $$ with the properties that $\gcd(a,b)=1$ and $p$ does not divide $a$ and $b$.
My question is, how can we verify that this representation is always possible?
My approach was to write $r :=\frac{c}{d}$ and use the unique prime factorization for $c$ and $d$. Then $e$ is just either $0$ if it does not appear in the prime factorization of $c$ or $d$ or in the other case, just factor out $p$ of the prime factorizations and rewrite $\frac{a}{b}$ without this $p$. But how can we now be sure that $\gcd(a,b)=1$ and $p$ does not divide $a$ and $b$?
Already, thank you very much for any help!