Bézout’s Identity
gcd is an integer combination of a and b.
Key idea
gcd is an integer combination of a and b.
In depth
For integers a, b, the equation ax + by = c has integer solutions exactly when gcd(a, b) divides c. The smallest positive value of ax + by is gcd(a, b).
Worked example
With 6x + 15y, the possible values are all multiples of 3.
Real-world uses
- Proving that combinations of two numbers can produce their greatest common divisor in algebra and coding
Common mistake
Assuming x and y are unique; there are infinitely many.
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 Bézout’s Identity?
gcd is an integer combination of a and b. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Bézout’s Identity used in real life?
Real-world examples: Proving that combinations of two numbers can produce their greatest common divisor in algebra and coding.
What should I learn before Bézout’s Identity?
Make sure you are comfortable with Extended Euclidean Algorithm, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Bézout’s Identity?
Assuming x and y are unique; there are infinitely many.
How can I practise Bézout’s Identity?
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 Bézout’s Identity?
Move on to Linear Diophantine Equations 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.