Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example
$\frac{10}{9}=\frac{2!5!}{3!3!3!}$.
It clearly suffices to prove that all prime numbers $p=\frac{p}{1}$ can be represented in this form.
We prove this by strong induction. For $p=2$ we can just write $2=2!$.
Suppose it is true for all primes less than $p$. Since $(p-1)!$ is a product of such primes, it can be written in the desired form. But then $p=\frac{p!}{(p-1)!}$ can also be written in this form. Done.