Euler’s Totient Function
Count integers coprime to n.
Key idea
Count integers coprime to n.
In depth
φ(n) counts integers from 1 to n coprime to n. For prime p, φ(p) = p − 1; φ(pᵏ) = pᵏ − pᵏ⁻¹; and φ is multiplicative for coprime arguments. Generally φ(n) = n Π(1 − 1/p).
Worked example
φ(12) = 12 × (1/2) × (2/3) = 4: the numbers 1, 5, 7, 11.
Formula
Real-world uses
- Counting numbers coprime to n
- The key quantity behind RSA encryption
Common mistake
Applying multiplicativity to numbers that share factors.
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 Totient Function?
Count integers coprime to n. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Euler’s Totient Function?
The key formula is φ(n) = n∏(1−1/p).
Where is Euler’s Totient Function used in real life?
Real-world examples: Counting numbers coprime to n; The key quantity behind RSA encryption.
What should I learn before Euler’s Totient Function?
Make sure you are comfortable with Fermat’s Little Theorem, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Euler’s Totient Function?
Applying multiplicativity to numbers that share factors.
How can I practise Euler’s Totient Function?
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 Totient Function?
Move on to Euler’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.