Math 2.0

Number theory Theorem

Product formula for the totient function

Builds on Euler's totient functionPrime number

Statement

Let n≥2n \ge 2 be an integer. Suppose that p1,…,pmp_1, \dots, p_m are the distinct prime numbers that divide nn. Then φ(n)=n∏i=1mpi−1pi,\varphi(n) = n \prod_{i=1}^{m} \frac{p_i - 1}{p_i}, where φ\varphi is Euler's totient function.

Remarks

Each factor equals 1−1/pi1 - 1/p_i, so only the set of primes dividing nn matters, not their exponents. For example, 12=22⋅312 = 2^2 \cdot 3 gives φ(12)=12⋅12⋅23=4\varphi(12) = 12 \cdot \frac{1}{2} \cdot \frac{2}{3} = 4.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionPrime number, definition, no evidence levelPrime numberdefinitionCoprime integers, definition, no evidence levelCoprime integersdefinitionEuler's totient function, definition, no evidence levelEuler's totient funct…definitionProduct formula for the totient function, statement, no evidence levelProduct formula for t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Greatest common divisor definition
  3. Prime number definition
  4. Coprime integers definition
  5. Euler's totient function definition
  6. Product formula for the totient function statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_inclusion-exclusion_euler-phi.ptx#L43, source/s_inclusion-exclusion_euler-phi.ptx:43
About this page
TagZCC0E1FW8P
URIm2://ZCC0E1FW8P
Kindstatement, theorem
WritingW1 stated
MSC11A25
StatusThis copy of the site has no build data. Run m2 status ZCC0E1FW8P.
UsesEuler's totient function, Prime number
Changed4 November 2026

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