Math 2.0

Number theory Corollary

The order modulo p divides p − 1

Builds on Multiplicative orderFermat's little theorem

Statement

Let pp be a prime number, and let aa be an integer with p∤ap \nmid a. Then the multiplicative order of aa modulo pp divides p−1p - 1.

Remarks

The proof: ap−1≡1(modp)a^{p-1} \equiv 1 \pmod p by Fermat's little theorem, and each exponent mm with am≡1a^m \equiv 1 is a multiple of the order. The order is equal to p−1p - 1 exactly when aa is a primitive root modulo pp. For example, modulo 1313, the order of 33 is 33 and the order of 22 is 1212. Both divide 1212.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionPrime number, definition, no evidence levelPrime numberdefinitionCoprime integers, definition, no evidence levelCoprime integersdefinitionFermat's little theorem, statement, no evidence levelFermat's little theor…statementEuler's totient function, definition, no evidence levelEuler's totient funct…definitionEuler's theorem, statement, no evidence levelEuler's theoremstatementMultiplicative order, definition, no evidence levelMultiplicative orderdefinitionThe order modulo p divides p − 1, statement, no evidence levelThe order modulo p di…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. Prime number definition
  5. Coprime integers definition
  6. Fermat's little theorem statement
  7. Euler's totient function definition
  8. Euler's theorem statement
  9. Multiplicative order definition
  10. The order modulo p divides p − 1 statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Multiplicative_order, Properties
About this page
Tag0Y4B7P0GS8
URIm2://0Y4B7P0GS8
Kindstatement, corollary
WritingW0 stub
MSC11A07
StatusThis copy of the site has no build data. Run m2 status 0Y4B7P0GS8.
UsesMultiplicative order, Fermat's little theorem

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