Exercise · ZW-BEZOUT-S-IDENTITY-005 · ★★☆ Intermediate
Which statement about the set \( \{ax+by : x,y\in\mathbb{Z}\} \) is correct?
Choose one option.
Show answer
- D. It contains exactly the multiples of \( \gcd(a,b) \)
Explanation
This set is exactly the set of multiples of the greatest common divisor. The option about multiples of the least common multiple is a frequent error.
Review the concept: Bézout's identity
Bézout's identity states that, for integers \( a \) and \( b \) not both zero, there exist integers \( x \) and \( y \) such that \( ax+by=\gcd(a,b) \).
\[ax+by=\gcd(a,b) \text{ for some } x,y\in\mathbb{Z}\]
More practice problems: bézout's identity
- ZW-BEZOUT-S-IDENTITY-001 State Bézout's identity for two integers a and b .
- ZW-BEZOUT-S-IDENTITY-002 What is the smallest positive integer that can be written as 12x+18y with x,y∈ℤ ?
- ZW-BEZOUT-S-IDENTITY-003 Give the greatest common divisor of 35 and 14 to use in Bézout's identity.
- ZW-BEZOUT-S-IDENTITY-004 Does Bézout's identity say that ax+by can equal any integer when a and b are coprime?
- ZW-BEZOUT-S-IDENTITY-006 If gcd(a,b)=d , which of these equations has an integer solution x,y for every such a and b ?
- ZW-BEZOUT-S-IDENTITY-007 For a=30 and b=12 , which pair (x,y) satisfies 30x+12y=6 ?
- ZW-BEZOUT-S-IDENTITY-008 Can 8x+12y equal 2 for some integers x,y ?
- ZW-BEZOUT-S-IDENTITY-009 For a=252 and b=198 , with gcd=18 , which pair (x,y) satisfies 252x+198y=18 ?