Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number that can be expressed as $$ N= \prod_{j=1}^{n} j^{x_j} $$ where $x_1$, $\ldots$, $x_n$ are non-negative real numbers with $\sum_{j}x_j \in {\Bbb N}$. Does there necessarily exist a representation of $N$ as $$ N=\prod_{j=1}^{n} j^{a_j} $$ where $a_1$, $\ldots$, $a_n$ are non-negative integers with $\sum_{j} a_j=\sum_j x_j$?

(Note: the condition that $\sum_j x_j \in {\Bbb N}$ is necessary, else as Christian Elsholtz kindly pointed out one has the counterexample $1^0\times 2^{1/2}\times 3^0\times 4^{1/4}=2$.)

This problem arose in connection with the recent MO question: How many different numbers can be obtained as product of first $n$ natural numbers? If the problem posed here has a positive answer, then it would follow (in the notation of the linked question) that $P(m,n)$ is the Erhart polynomial (in $m$) of a certain convex polytope ${\mathcal C}(n)$.

The elementary problem given above may be thought of as a question on lattice points in polytopes (in ${\Bbb R}^{\pi(n)}$) by considering the exponents in the prime factorizations of numbers below $n$, and it could be the case that a related question has already been investigated in that context. Hence I have added the algebraic geometry and convex polytopes tags to see if the experts there already know the answer.