Exercise · ZW-EXTENDED-EUCLIDEAN-ALGORITHM-003 · ★☆☆ Basic
If \(\gcd(a,b)=d\), what relation do \(a\), \(b\) and \(d\) satisfy in Bézout's identity?
Choose one option.
Show answer
- B. \(d=ax+by\) for some integers \(x,y\)
Explanation
Bézout's identity ensures an integer linear combination of \(a\) and \(b\) equal to \(d\). The product option confuses it with a multiplicative relation that does not hold in general.
Review the concept: Extended Euclidean algorithm
The extended Euclidean algorithm computes \(d=\gcd(a,b)\) and integers \(x\), \(y\) such that \(ax+by=d\). It is the Euclidean algorithm that also tracks the Bézout coefficients.
\[r_k=r_{k-2}-q_{k-1}r_{k-1}\\s_k=s_{k-2}-q_{k-1}s_{k-1}\\t_k=t_{k-2}-q_{k-1}t_{k-1}\]
More practice problems: extended euclidean algorithm
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-001 What does the extended Euclidean algorithm guarantee at termination?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-002 What are the initial values of s_-1 and t_-1?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-004 In the step 46=4· 10+6 of the division r_k-2=q_k-1r_k-1+r_k, what is q_k-1?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-005 What is gcd(240,46) according to the algorithm?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-006 For a=30 and b=12, with 6=30-2· 12, what value of y satisfies 30x+12y=6 with x=1?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-007 With a=35 and b=15, the algorithm gives gcd=5. Which is a valid pair (x,y) with 35x+15y=5?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-008 For a=17 and b=5, what is the coefficient y in the identity 17x+5y=1?
- ZW-EXTENDED-EUCLIDEAN-ALGORITHM-009 If the extended algorithm gives s_k and t_k with r_k=as_k+bt_k, what must hold at the end of the pr…
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