Greatest Common Divisor and Primes

Question

Let $a = r^2 s^2$ for distinct primes $r$ and $s$ and let $x > 1$, $\in \mathbb{Z}$. $x=y r^k s^l$, $k, l$ are non-negative integers and $y$ is $1$ or a product of primes other than $r$ and $s$.

Prove that $r|x$ or $s|x$ $\iff$ $gcd(x, a) \neq 1 $

Answer 1

the $\implies$ is immediate since then either $r$ or $s$ will be a divisor of the $gcd(a,x)$.

for the reverse

if $gcd(a,x) \neq 1$ then $\exists p \in \mathbb{Z}$ such that $z|a$ and $z|x$

but since $a= r^2s^2 \implies r|z \ or \ s|z$ (since $r \ and \ s$ are primes)

thus $r|x$ or $s|x$ .