Exercise · ZW-BEZOUT-S-IDENTITY-004 · ★★☆ Intermediate
Does Bézout's identity say that \( ax+by \) can equal any integer when \( a \) and \( b \) are coprime?
Choose one option.
A No, it only guarantees that \( ax+by=1 \) has an integer solution B Yes, provided the coefficients are positive C Yes, because coprime means that \( a \) and \( b \) are prime D No, because the identity does not apply to coprime numbers Show answer A. No, it only guarantees that \( ax+by=1 \) has an integer solution← Back to the question
Explanation If \( \gcd(a,b)=1 \), the identity gives \( ax+by=1 \), not every integer. The option requiring positive coefficients is false, because they may be negative.
Bézout's identity states that, for integers \( a \) and \( b \) not both zero, there exist integers \( x \) and \( y \) such that \( ax+by=\gcd(a,b) \).
\[ax+by=\gcd(a,b) \text{ for some } x,y\in\mathbb{Z}\]
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