TITLE: Properties of the prime numbers?
QUESTION [6 upvotes]: Consider the following exercise:

Let $P_1$ be the set of all primes, $\{2,3,5,7,\cdots\}$, and for each integer $n$, let $P_n$ be the set of all prime multiples of $n$, $\{2n,3n,5n,7n,\cdots\}$. Which of the following intersections is nonempty?
A $P_{1}\cap P_{23}$
B $P_{7}\cap P_{21}$
C $P_{12}\cap P_{20}$
D $P_{20}\cap P_{24}$
E $P_{5}\cap P_{25}$

Here are my questions:

When are the integer $m$ and $n$ such that $P_m\cap P_n\neq\emptyset $?

Are there any other methods to the solve the question in the exercise except than "trial and error"?

REPLY [8 votes]: 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 $\{60\}$