Linear Diophantine Equations
Integer solutions of ax + by = c.
Key idea
Integer solutions of ax + by = c.
In depth
ax + by = c has integer solutions iff gcd(a, b) | c. If (x₀, y₀) is one, all solutions are x = x₀ + (b/g)t, y = y₀ − (a/g)t. Find one by the extended Euclidean algorithm.
Worked example
3x + 5y = 1: (2, −1) works; general solution (2 + 5t, −1 − 3t).
Real-world uses
- Solving puzzles such as paying exactly £50 with £3 and £5 notes using whole numbers
Common mistake
Missing the general family and giving just one solution.
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 Linear Diophantine Equations?
Integer solutions of ax + by = c. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Linear Diophantine Equations used in real life?
Real-world examples: Solving puzzles such as paying exactly £50 with £3 and £5 notes using whole numbers.
What should I learn before Linear Diophantine Equations?
Make sure you are comfortable with Bézout’s Identity, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Linear Diophantine Equations?
Missing the general family and giving just one solution.
How can I practise Linear Diophantine Equations?
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 Linear Diophantine Equations?
Move on to Congruences 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.