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# Implications of Bézout's Theorem

NUMTH-2DRWUT

Let $a, b, c \in \mathbb{Z} \backslash \{ 0 \}$.

Which of the following is TRUE?

A

There exist $x_1, ..., x_6 \in\mathbb{Z}$ such that $a(x_1 + x_3) + b(x_2 + x_5) + c(x_4 + x_6) = \gcd(a, b) + \gcd(b, c) + \gcd(a, c).$

B

There exist $y_1, ..., y_6 \in\mathbb{Z}$ such that $(ay_1 + by_2)(ay_3 + cy_4)(by_5 + cy_6) = \gcd(a, b)\gcd(a,c)\gcd(b,c).$

C

There exist $z_1, z_2 \in \mathbb{N}$ such that $az_1 + az_2 = \gcd(a, a).$

D

There exist $w_1, w_2 \in \mathbb{R}$ such that $aw_1 + aw_2 = \gcd(a, a).$