Question 13
Let $q$ be a fixed positive rational number. Call number $x$ *charismatic* if there exist a positive integer $n$ and integers $\alpha_1, \alpha_2, \dots, \alpha_n$ such that
$$
x = (q+1)^{\alpha_1} \cdot (q+2)^{\alpha_2} \cdots (q+n)^{\alpha_n}.
$$
a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic.
b) Is it true for every $q$ that, for every charismatic number $x$, the number $x+1$ is charismatic too?
$$
x = (q+1)^{\alpha_1} \cdot (q+2)^{\alpha_2} \cdots (q+n)^{\alpha_n}.
$$
a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic.
b) Is it true for every $q$ that, for every charismatic number $x$, the number $x+1$ is charismatic too?