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Which of the following statements truthfully complete the following sentence? Select ALL that apply.

For $a_1, a_2, ..., a_n \in \mathbb{Z} \backslash \{ 0 \}$, there exist $x_1, x_2, ..., x_n \in \mathbb{Z}$ such that ...

A

$(a_1 x_1 + a_2 x_2)(a_3 x_3 + a_4 x_4) \cdots (a_{n - 1} x_{n - 1} + a_n x_n) = \gcd(a_1, a_2) \gcd(a_3, a_4) \cdots \gcd(a_{n - 1}, a_n)$

B

$\displaystyle\sum_{m = 1}^{n} a_m x_m =\gcd(a_1, a_2) + \gcd(a_3, a_4) + \cdots + \gcd(a_{n - 1}, a_n)$

C

$\displaystyle\sum_{m = 1}^{n} a_m x_m = \gcd(a_1, ..., a_n)$

D

$\displaystyle\sum_{m = 1}^{n} a_m x_m =\sqrt{\gcd(a_1, ..., a_n)}$

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