Properties of the prime numbers?

Consider $P_m \cap P_n$ where $m \neq n$

If $x \in P_m \cap P_n$, then be definition, $x = mp_m$ and $x = np_n$ for some prime $p_m$ and $p_n$.

Hence, we get that $\frac{m}{n} = \frac{p_n}{p_m}$

So the intersection of $P_m$ and $P_n$ is non-trivial iff the $\frac{m}{n}$ in the reduced form is a ratio of two primes.

In fact, if $\frac{m}{n}$ in the reduced form is a ratio of two primes, then the intersection has only one element namely $\{\text{lcm}(m,n)\}$

In your case, the intersection is non-trivial only for the case $P_{12} \cap P_{20}$ which is nothing but $\{60\}$

Some hint:

Suppose $x\in P_m\cap P_n$ that means that there are two prime numbers $p,q$ such that $x=pm=qn$.

• If $p=q$ then clearly $m=n$.
• Otherwise $p\mid qn$ therefore $p\mid n$, and just as well $q\mid m$. When does that happen?