Question 104
Let $n \geqslant 3$ be an integer. For each pair of prime numbers $p$ and $q$ such that $p\lt q \leqslant n$, Morgane has written the sum $p+q$ on the board. She then denotes by $\mathcal{P}(n)$ the product of all these sums. For example, $\mathcal{P}(5)=(2+3) \times (2+5) \times (3+5)=280$.
Find all values of $n \geqslant 3$ for which $n!$ divides $\mathcal{P}(n)$.
Note: If two sums $p+q$ formed from two different pairs are equal to each other, Morgane writes both of them. For example, if $n=13$, she does write both sums $3+13$ and $5+11$.
Find all values of $n \geqslant 3$ for which $n!$ divides $\mathcal{P}(n)$.
Note: If two sums $p+q$ formed from two different pairs are equal to each other, Morgane writes both of them. For example, if $n=13$, she does write both sums $3+13$ and $5+11$.