Difficult# Examining Modifications of Bézout's Identity

NUMTH-HPG3N5

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?