Extended Euclidean Algorithm
Find x, y with ax + by = gcd(a,b).
Key idea
Find x, y with ax + by = gcd(a,b).
In depth
It runs the Euclidean algorithm and works backwards to write gcd(a, b) = ax + by. This produces the coefficients (Bézout coefficients), which give modular inverses.
Worked example
gcd(240, 46) = 2 = 240(−9) + 46(47).
Formula
Real-world uses
- Finding modular inverses for encryption keys and solving ax + by = c problems
Common mistake
Making arithmetic mistakes when back-substituting; verify the final identity.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Extended Euclidean Algorithm?
Find x, y with ax + by = gcd(a,b). It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Extended Euclidean Algorithm?
The key formula is ax + by = gcd.
Where is Extended Euclidean Algorithm used in real life?
Real-world examples: Finding modular inverses for encryption keys and solving ax + by = c problems.
What should I learn before Extended Euclidean Algorithm?
Make sure you are comfortable with GCD and LCM, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Extended Euclidean Algorithm?
Making arithmetic mistakes when back-substituting; verify the final identity.
How can I practise Extended Euclidean Algorithm?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Extended Euclidean Algorithm?
Move on to Bézout’s Identity next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.