Euler Totient Function Calculator

What Is Euler's Totient Function?

Euler's totient function $\varphi(n)$ counts how many integers from 1 to $n$ are coprime to $n$. Two integers are coprime when their greatest common divisor is 1.

Product Formula

If $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, then:

Key Properties

• For prime $p$: $\varphi(p) = p - 1$
• For prime power $p^k$: $\varphi(p^k) = p^k - p^{k-1}$
• If $\gcd(m,n)=1$, then $\varphi(mn)=\varphi(m)\varphi(n)$
• Euler's theorem: $a^{\varphi(n)} \equiv 1 \pmod{n}$ when $\gcd(a,n)=1$

The totient function is central to RSA cryptography because private-key operations rely on $\varphi(n)$ for semiprime moduli. Related tools: use the Prime Factors Calculator and GCF Calculator.