Euler’s Theorem
a^φ(n) ≡ 1 mod n for coprime a, n.
Key idea
a^φ(n) ≡ 1 mod n for coprime a, n.
In depth
If gcd(a, n) = 1 then a^φ(n) ≡ 1 (mod n). It generalises Fermat’s little theorem to any modulus and enables fast modular exponentiation and RSA.
Worked example
φ(10) = 4, so 3⁴ = 81 ≡ 1 (mod 10).
Formula
Real-world uses
- Reducing large exponents in cryptography and checking coding schemes
Common mistake
Using it when a and n are not coprime.
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 Euler’s Theorem?
a^φ(n) ≡ 1 mod n for coprime a, n. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Euler’s Theorem?
The key formula is a^φ(n) ≡ 1 (mod n).
Where is Euler’s Theorem used in real life?
Real-world examples: Reducing large exponents in cryptography and checking coding schemes.
What should I learn before Euler’s Theorem?
Make sure you are comfortable with Euler’s Totient Function, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Euler’s Theorem?
Using it when a and n are not coprime.
How can I practise Euler’s Theorem?
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 Euler’s Theorem?
Move on to Wilson’s Theorem 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.