TITLE: Can the product of $n$ factorials be $n$ factorial?
QUESTION [6 upvotes]: Are there any solutions to the equation $a_1!\cdot a_2!\cdots a_n!=n!$ with all variables being integers greater than or equal to $2$?

REPLY [7 votes]: Legendre's formula implies that $n!$ is not a multiple of $2^n$: the power of $2$ in its prime factorization is $\sum_{k=1}^\infty \lfloor n/2^k \rfloor < n$.
On the other hand, the left-hand side is a multiple of $2^n$.