Euler’s Totient Function (φ(n)) – Complete Guide with Examples
Euler’s Totient Function, denoted as φ(n), is one of the most important functions in number theory. It counts the number of positive integers less than or equal to n that are relatively prime to n. This concept is widely used in cryptography, modular arithmetic, and competitive exams like CBSE Class 10.
What is Euler’s Totient Function?
The Euler Totient Function φ(n) gives the count of numbers from 1 to n that do not share any common factor with n other than 1.
Geometric Visualization
Circle representing numbers around n showing coprime distribution
Formula for Euler Totient Function
If n is a prime number:
If n = p × q (product of two primes):
General Formula:
Important Properties
- φ(p) = p − 1 for prime p
- φ(mn) = φ(m) × φ(n) if m and n are coprime
- φ(n) is always even for n > 2
- Used in RSA encryption
Solved Problems
Solution: 9 = 3² φ(9) = 9 × (1 – 1/3) = 9 × (2/3) = 6
Solution: 15 = 3 × 5 φ(15) = 15 × (1 – 1/3)(1 – 1/5) = 15 × (2/3)(4/5) = 8
Solution: 21 = 3 × 7 φ(21) = 21 × (2/3)(6/7) = 12
Solution: 25 = 5² φ(25) = 25 × (1 – 1/5) = 20
Solution: 36 = 2² × 3² φ(36) = 36 × (1 – 1/2)(1 – 1/3) = 36 × (1/2)(2/3) = 12
Applications of Euler Totient Function
- Cryptography (RSA Algorithm)
- Modular Arithmetic Problems
- Finding multiplicative inverses
- Computer Science & Security Systems
Why is it Important for CBSE Students?
Euler’s Totient Function is frequently asked in board exams and helps build a strong base in number theory. Understanding φ(n) makes solving LCM, HCF, and modular arithmetic problems easier.
Conclusion
Euler’s Totient Function is not just a formula but a powerful mathematical tool. Mastering it can significantly improve your problem-solving skills in mathematics.