Find the solution to each of distinct positive x_i 's

Question

$\text{Solve the equation:}$ $$ 1+x_1+2x_1x_2+\cdots+(n-1)x_1x_2...x_{n-1}=x_1x_2x_3\cdots x_n$$ in distinct positive integers $x_1,x_2,\cdots,x_n$

I did some divisibility stuffs and can understand that $x_i=i~ \forall~ 1\leq i\leq n$ But I can't prove that. I am getting $x_i \mid i!$ using telescoping sums..

Anyway, I can't proceed.

Answer 1

hint

If $x_1>1$. Let prime $p$ divide $x_1$, then right side is divisible by $p$ and on the left side all terms starting from the second term onwards are divisible by $p$. Thus $p$ has to divide $1$, but then $x_1=1$. Now do the same with $x_2$.