Question 180
Let $\mathbb{P}$ be the set of all prime numbers and let $M$ be a subset of $\mathbb{P}$ having at least three elements. Suppose that for every integer $k \geq 1$ and for every subset $A=\{p_{1}, p_{2}, \ldots, p_{k}\}$ of $M$ with $A \neq M$, all prime factors of the number $p_{1} \cdot p_{2} \cdot \ldots \cdot p_{k}-1$ lie in $M$. Show that $M=\mathbb{P}$.