Euler Totient Calculator – φ(n) Coprimes

Euler’s totient counts how many integers from 1 to n share no factor with n. For 12 = 2²×3: φ(12) = 12×(1−½)×(1−⅓) = 12×½×⅔ = 4 — the coprimes 1, 5, 7, 11, which the calculator lists. For prime powers, φ(9) = 9−3 = 6 (removing multiples of 3). Primes are maximal: φ(13) = 12 since nothing below shares a factor. The function is multiplicative on coprime factors: φ(10) = φ(2)×φ(5) = 1×4 = 4. The tool factorizes n, applies the product formula with each (1−1/p) shown, lists the coprimes for small n, and displays Euler’s theorem: a^φ(n) ≡ 1 (mod n) for gcd(a,n)=1 — the engine behind RSA, where ed ≡ 1 (mod φ(n)) pairs keys. For n = 1, φ(1) = 1 by convention.

1. 1

   Enter n

   Any positive integer

2. 2

   Factorize

   Distinct primes of n are extracted (12 → 2, 3)

3. 3

   Apply the product formula

   n·Π(1−1/p) with each factor evaluated

4. 4

   List the coprimes

   Small n show the actual coprime set (1, 5, 7, 11)

• 1

  φ(p) = p−1 for every prime p

• 2

  Product formula: φ(12) = 12·(1−½)·(1−⅓) = 4

• 3

  φ is multiplicative on coprime inputs

• Computing φ(pᵏ) as pᵏ−1 instead of pᵏ−pᵏ⁻¹

• Using all prime factors with multiplicity — only distinct primes enter the formula

• Assuming φ is additive: φ(m+n) ≠ φ(m)+φ(n) in general

• Forgetting φ(1) = 1