This is exercise 16 from section 2, from Elementary Number Theory by Underwood Dudley. It states that letters $p$ and $q$ are reserved for primes.
This is my proof, I'm not sure if it's correct:
Since $p | n$ and $q | n,$ and both are primes then $p \leq \sqrt{n}$ and $q \leq \sqrt{n}$
Multiplying both inequalities
$pq \leq n$
$1 \leq \frac{n}{pq}$
Which means that $\frac{n}{pq}$ can be $1$, thus it must be false.
Is this correct?
If it's correct, I would love to see alternative proofs.
The claim is false as stated. For a concrete example, if $n=60$, $p=3$, $q=5$, then
$$\sqrt[4]{60}\approx 2.783 < 3,\ 5,$$
but $\frac{60}{3\cdot 5}=4$ is not prime (as MJD noted, $n=6,p=2,q=3$ is also a counterexample).
In your proof, the statement $p,q\leq \sqrt{n}$ is wrong; $p$ and $q$ cannot both be greater than $\sqrt{n}$ (since that would mean $pq>n$), but it is not necessarily true that both are less (consider $n=12q$ and $p=3$ for some large $q$, for example).