Math 2.0

Number theory Proposition

Number of invertible residues modulo n

Builds on Integers modulo nEuler's totient functionExistence of inverses modulo n

Statement

Let nn be a positive integer. The number of invertible elements of the ring ℤ/nℤ\mathbb{Z}/n\mathbb{Z} of the integers modulo n is φ(n)\varphi(n), where φ\varphi is Euler's totient function.

Remarks

By the existence of inverses modulo n, the class a‾\overline{a} is invertible if and only if gcd⁡(a,n)=1\gcd(a, n) = 1. The classes of 1,…,n1, \dots, n that are coprime to nn give φ(n)\varphi(n) classes. The invertible elements form a group under multiplication, the group of units (ℤ/nℤ)×(\mathbb{Z}/n\mathbb{Z})^\times. It has order φ(n)\varphi(n). For example, (ℤ/8ℤ)×={1‾,3‾,5‾,7‾}(\mathbb{Z}/8\mathbb{Z})^\times = \{ \overline{1}, \overline{3}, \overline{5}, \overline{7} \} has φ(8)=4\varphi(8) = 4 elements.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCongruence is compatible with addition, statement, no evidence levelCongruence is compati…statementCongruence is compatible with multiplication, statement, no evidence levelCongruence is compati…statementModular multiplicative inverse, definition, no evidence levelModular multiplicativ…definitionResidue class, definition, no evidence levelResidue classdefinitionBézout's identity, statement, no evidence levelBézout's identitystatementCoprime integers, definition, no evidence levelCoprime integersdefinitionIntegers modulo n, definition, no evidence levelIntegers modulo ndefinitionExistence of inverses modulo n, statement, no evidence levelExistence of inverses…statementEuler's totient function, definition, no evidence levelEuler's totient funct…definitionNumber of invertible residues modulo n, statement, no evidence levelNumber of invertible …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. Congruence modulo n definition
  3. Greatest common divisor definition
  4. Congruence is compatible with addition statement
  5. Congruence is compatible with multiplication statement
  6. Modular multiplicative inverse definition
  7. Residue class definition
  8. Bézout's identity statement
  9. Coprime integers definition
  10. Integers modulo n definition
  11. Existence of inverses modulo n statement
  12. Euler's totient function definition
  13. Number of invertible residues modulo n statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler%27s_totient_function, Euler's theorem
About this page
Tag06RJWZCG95
URIm2://06RJWZCG95
Kindstatement, proposition
WritingW0 stub
MSC11A07
StatusThis copy of the site has no build data. Run m2 status 06RJWZCG95.
UsesIntegers modulo n, Euler's totient function, Existence of inverses modulo n

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