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"?