Question 56
Let $n \geq 3$ be a positive integer and $M$ be the set of the first $n$ prime numbers. For every nonempty subset $X$ of $M$ denote by $P(X)$ the product of the elements of $X$. Let $N$ be a set of fractions of the form $\frac{P(A)}{P(B)}$, where $A \subset M$, $B \subset M$, $A \cap B = \varnothing$ such that the product of any 7 elements of $N$ is an integer. What is the maximum possible cardinality of $N$?