Euler's totient: φ(n) = count of 1≤k≤n with gcd(k,n)=1. Formula: φ(n) = n·Π(1−1/p) over prime divisors p of n. Properties: φ(p)=p−1, φ(pᵏ)=pᵏ−pᵏ⁻¹, multiplicative: φ(mn)=φ(m)φ(n) when gcd(m,n)=1. Euler's theorem: aᶲ⁽ⁿ⁾≡1 (mod n).
How to Use
Enter n:
- n: Positive integer
- φ(n): Totient value
- Factors: Prime factorization
Properties
- Multiplicative: φ(mn)=φ(m)φ(n) when coprime
- Σ_{d|n} φ(d) = n
- φ(p) = p−1 (prime)
- φ(2ⁿ) = 2ⁿ⁻¹
Applications
- RSA key generation
- Primitive roots
- Multiplicative group structure
- Counting in cyclic groups
Step-by-Step Instructions
- 1Enter n.
- 2Get φ(n).
- 3View prime factors.
- 4See coprime list.
- 5Verify Euler's theorem.