My friend asked me this question. I would appreciate your understanding even if my English is not perfect.
For any prime number ${p}$, let's define a set ${P}$ that consists of all multiples of ${p}$. Also, let's define a set ${S}$ that contains all multiples of each prime number smaller than ${p}$. If we consider the set of all natural numbers as ${N}$, how do the sizes of the following two sets compare?
$N \cap S^C$ and $P$
For example, if ${p=7}$: $$P= \{7,14,21,28,35,42,… \}$$ $$S= \{2,3,4,5,6,8,9,10,12,14,15,16,18,20,21,… \}$$ $$N \cap S^{C}=\{1,7,11,13,17,19,… \}$$ I believe I can establish a one-to-one correspondence between $N \cap S^{C}$ and ${P}$, so I think they have the same cardinality as ${N}$.