Euler's Totient Function Calculator

About this calculator

Euler's totient φ(n) counts how many integers from 1 to n share no common factor with n, and is computed from n's prime factorization as φ(n) = n × ∏(1 − 1/p) over each distinct prime p dividing n. For a prime p itself, φ(p) = p − 1 since every smaller positive integer is automatically coprime to it.

The totient function is the mathematical engine behind RSA encryption: key generation multiplies two large primes p and q to get n = pq, then uses φ(n) = (p−1)(q−1) to compute a valid public/private exponent pair. It also underlies Euler's theorem in number theory, used to find modular inverses and reduce large exponents in cryptographic and competitive-programming calculations.