OIMU 2009 Problem 6

Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials

$f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$

have integer coefficients.

Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$.

Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1$