Euler's Totient (Phi) Calculator
Euler's totient φ(n) counts how many integers from 1 to n share no common factor with n other than 1.
Count of integers in 1..n coprime to 36
12
coprimeCoprime to n
33.3%
Sharing a factor
66.7%
- 1
Distinct prime factors of n
2, 3 - 2
Apply product formula φ(n) = n × Π(1 − 1/p)
36 × (1 − 1/2) × (1 − 1/3) = 12
How does this calculator work?
Euler's totient φ(n) counts the integers from 1 to n that are coprime to n. Factor n into its distinct primes p and compute φ(n) = n · Π(1 − 1/p). The coprime ratio φ(n)/n equals that product and gives the chance a random integer below n is coprime to n.
Formula
How this is calculated
Enter a single positive integer n (n ≥ 1). The calculator factors n by trial division, finding each distinct prime p that divides it. Repeated prime factors are collapsed, so only the set of distinct primes matters for the totient.
Starting from φ = n, the result is multiplied by (1 − 1/p) for every distinct prime p, which is implemented as φ = φ − φ/p to avoid floating-point drift; the final value is rounded to the nearest integer. This is the product formula φ(n) = n · Π(1 − 1/p), which holds because the totient is multiplicative over coprime prime powers. The coprime ratio φ(n)/n equals that same product and represents the probability that a random integer in 1..n is coprime to n.
Edge cases: φ(1) = 1 by convention, and a prime n returns φ(n) = n − 1 with itself as the only prime factor. Inputs must be whole positive numbers; non-integers, values below 1, or extremely large n (above 10^12) are rejected. Outputs are pure counts and have no physical units.
Frequently asked questions
By convention φ(1) = 1, since 1 is counted as coprime to itself and there are no prime factors to reduce the count.
A prime p shares no factor with any of 1, 2, …, p − 1, so all p − 1 of those integers are coprime to it, leaving only p itself excluded.
No. Only the distinct primes dividing n appear in the product Π(1 − 1/p); their multiplicities are absorbed into the leading factor n.
Also known as
TG we-Calculate Editorial Team. (2026). Euler's Totient (Phi) Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/euler-totient-calculator
TG we-Calculate Editorial Team. "Euler's Totient (Phi) Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/euler-totient-calculator.
TG we-Calculate Editorial Team, "Euler's Totient (Phi) Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/euler-totient-calculator
@misc{wecalculate_euler_totient_calculator, title = {Euler's Totient (Phi) Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/euler-totient-calculator}}, year = {2026}, note = {TG we-Calculate} }
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