Let $x_1, x_2,\dots, x_n$ be an arrangement of the numbers $1, 2,\dots, n$. Show that if n is odd, then $(x_1 − 1)(x_2 − 2)\dots(x_n − n)$ is even.
About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$