Math 2.0

Number theory Proposition

Units of Z/nZ are the residues coprime to n

Builds on Group of units of a monoidGroup of units of Z/nZ

Statement

Let n≥1n \geq 1 and aa be integers. The residue class [a][a] is a unit of the ring ℤ/nℤ\mathbb{Z}/n\mathbb{Z} if and only if gcd⁡(a,n)=1\gcd(a, n) = 1.

Remarks

If gcd⁡(a,n)=1\gcd(a, n) = 1, then Bézout's lemma gives integers xx and yy with ax+ny=1a x + n y = 1. So [a][x]=[1][a][x] = [1]. If [a][x]=[1][a][x] = [1], then ax−1a x - 1 is a multiple of nn, so each common divisor of aa and nn divides 11. The condition does not depend on the representative, because a≡b(modn)a \equiv b \pmod{n} implies gcd⁡(a,n)=gcd⁡(b,n)\gcd(a, n) = \gcd(b, n). For n=12n = 12, the units are [1][1], [5][5], [7][7] and [11][11].

Reading path

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup of units of Z/nZ, object, no evidence levelGroup of units of Z/nZobjectAssociativity, definition, no evidence levelAssociativitydefinitionIdentity element, definition, no evidence levelIdentity elementdefinitionSemigroup, definition, no evidence levelSemigroupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionMonoid, definition, no evidence levelMonoiddefinitionGroup of units of a monoid, definition, no evidence levelGroup of units of a m…definitionUnits of Z/nZ are the residues coprime to n, statement, no evidence levelUnits of Z/nZ are the…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. Binary operation definition
  2. Group of units of Z/nZ object
  3. Associativity definition
  4. Identity element definition
  5. Semigroup definition
  6. Inverse element definition
  7. Monoid definition
  8. Group of units of a monoid definition
  9. Units of Z/nZ are the residues coprime to n statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n, Group axioms
About this page
Tag0GXNHEHQNA
URIm2://0GXNHEHQNA
Kindstatement, proposition
WritingW0 stub
MSC11A07, 20K01
StatusThis copy of the site has no build data. Run m2 status 0GXNHEHQNA.
UsesGroup of units of a monoid, Group of units of Z/nZ

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