Number Theory

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More Implications of Bézout's Identity

NUMTH-UCSJMW

For $a, b \in \mathbb{Z} \backslash \{0\}$, which of the following statements are true?

Select ALL that apply.

A

There exists $x_1, x_2, x_3, x_4 \in \mathbb{Z}$ such that $(ax_1 + bx_2)^{ax_3 + bx_4} = \gcd(a, b)^{\gcd(a, b)}$.

B

There exists $x, y \in \mathbb{Z}$ such that $a^2 x^2 + 2abxy + b^2 y^2 = (\gcd(a, b))^2$.

C

There exists $w, x, y, z \in \mathbb{Z}$ such that $a(w - z) + b(x - y) = 0$.

D

There exists $x, y \in \mathbb{Z}$ such that $e^{ax} e^{by} = e^{\gcd(a, b)}$.