Euler's Totient Calculator

Euler's totient function phi(n), also written as the Greek letter phi of n, counts how many integers from 1 to n share no common factor with n other than 1 (the integers coprime to n). For example, phi(10) is 4 because only 1, 3, 7, and 9 are coprime to 10. By convention phi(1) equals 1. It is one of the most important functions in number theory.

Write n as a product of prime powers, n = p1^a1 times p2^a2 and so on. Then Euler's product gives phi(n) = n times the product over each distinct prime p of (1 - 1/p). An equivalent integer form is the product of p^(a-1) times (p - 1) over the distinct primes. This calculator factors n first, then applies the integer form, so the result stays exact.

Take n = 36, which factors as 2^2 times 3^2, so the distinct primes are 2 and 3. Euler's product gives phi(36) = 36 times (1 - 1/2) times (1 - 1/3) = 36 times 1/2 times 2/3 = 12. So 12 of the integers from 1 to 36 are coprime to 36, and the coprime density phi(n)/n is 12/36, or about 33 percent.

When p is prime, its only divisors are 1 and p itself, so every integer from 1 up to p - 1 shares no factor with p other than 1. That makes all p - 1 of them coprime to p, giving phi(p) = p - 1. This calculator flags when your input is prime and confirms that phi(n) equals n - 1 in that case.

Two integers are coprime (relatively prime) when their greatest common divisor is 1, meaning they share no prime factor. Euler's totient is multiplicative for coprime arguments: if gcd(m, n) is 1, then phi(m times n) equals phi(m) times phi(n). That property is exactly why the totient can be built up from the prime-power factors of n.

The totient underpins Euler's theorem, which states that a raised to the power phi(n) is congruent to 1 modulo n whenever a and n are coprime. That result is the foundation of RSA public-key cryptography, where phi of the modulus determines the decryption exponent. The totient also appears in modular arithmetic, the order of the group of units modulo n, and cyclic-group theory.