Recall that Bézout's Identity states that for $a, b \in \mathbb{Z} \backslash \{0 \}$, there exist $x, y \in \mathbb{Z}$ such that $ax + by = \gcd(a, b)$.

Which of the following modifications of Bézout's Identity are true?

A

For $a \in \mathbb{Z}$ and $b \in \mathbb{Z} \backslash \{0 \}$, there exists $x, y \in \mathbb{Z}$ such that $ax + by = \gcd(a, b)$.

B

For $a, b \in \mathbb{Z}$, there exists $x, y \in \mathbb{Z}$ such that $ax + by = \gcd(a, b)$.

C

For $a, b \in \mathbb{Z} \backslash \{0\}$, we have for every $x, y \in \mathbb{Z}$, that $ax + by = \gcd(a, b)$.

D

For $a, b \in \mathbb{N}$, there exist $x, y \in \mathbb{N}$ such that $ax + by = \gcd(a, b)$.