Prove that every divisor of a Hurwitz polynomial is a Hurwitz polynomial.

Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients whose constant term equals 1. Show that there exists an integer $b \geq 1$ such that $Q(bx)$ has integer coefficients.

Show that Theorem 1 is equivalent to the following statement:
Let $f(x) \in \mathbf{Q}\llbracket x \rrbracket$ be an exponential polynomial such that $f(1) = \sum_{i=1}^{s} P_i(1) e^{c_i}$ vanishes. Then $f(x)/(x-1)$ is still an exponential polynomial.
(An exponential polynomial is any power series with rational coefficients of the form $f(x) = \sum_{i=1}^{s} P_i(x) e^{c_i x}$, where $c_1, \ldots, c_s \in \mathbf{Q}$ are rationals and $P_1, \ldots, P_s \in \mathbf{Q}[x]$ are polynomials.)