MultiplicativeOrder [k,n]
gives the multiplicative order of k modulo n, defined as the smallest integer such that .
MultiplicativeOrder [k,n,{r1,r2,…}]
gives the generalized multiplicative order of k modulo n, defined as the smallest integer such that for some .
MultiplicativeOrder
MultiplicativeOrder [k,n]
gives the multiplicative order of k modulo n, defined as the smallest integer such that .
MultiplicativeOrder [k,n,{r1,r2,…}]
gives the generalized multiplicative order of k modulo n, defined as the smallest integer such that for some .
Details
- MultiplicativeOrder is also known as modulo order or haupt‐exponent.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- Typically used in modular arithmetic and cryptography.
- MultiplicativeOrder [k,n] gives the smallest positive integer m such that the remainder when dividing km by n is equal to 1.
- MultiplicativeOrder returns unevaluated if there is no integer satisfying the necessary conditions.
- For a FiniteFieldElement object a, MultiplicativeOrder [a] gives the multiplicative order of a, defined as the smallest positive integer m such that is the multiplicative identity of the finite field.
Examples
open all close allBasic Examples (2)
The multiplicative order of 5 modulo 8:
Plot the sequence with a fixed modulus:
Plot the sequence, varying the modulus:
Scope (7)
Numerical Evaluation (5)
Compute using integers:
Generalized multiplicative order:
Compute using large numbers:
Multiplicative order of finite field elements:
TraditionalForm formatting:
Symbolic Manipulation (2)
Use Solve to find solutions of equations:
Use FindInstance to find solutions:
Applications (9)
Basic Applications (5)
Find all primitive roots modulo 43:
A rational number has a digit cycle of length if is prime and 10 is a primitive root for :
Compute MultiplicativeOrder using NestWhileList :
Count number of possible multiplicative orders modulo a given prime number:
The number of divisors of where is prime:
These are in fact the same list:
Number Theory (4)
The repetition period in Rule for odd divides q[n]:
The digits of in base repeat with period :
The function digitCycleLength gives the digit period for any rational number in base :
This shows that the decimal representation of in base 10 repeats every 3 digits:
Build an RSA-like toy encryption scheme:
Perform a cycling attack. One of the outputs will be the plaintext:
Properties & Relations (5)
The multiplicative order of a primitive root modulo n is EulerPhi [n]:
EulerPhi divides MultiplicativeOrder :
The result is always positive:
Find the smallest integer such that ≡ 2, 3, or 4 mod 7:
Find which of the remainders satisfies :
Solve the discrete log problem with :
Possible Issues (1)
For nonzero integers k and n, MultiplicativeOrder [k,n] exists if and only if k and n are coprime:
However, 10 and 22 are not coprime:
Interactive Examples (1)
MultiplicativeOrder of each integer below a given prime number:
Neat Examples (2)
Visualize when a number has multiplicative order modulo 12:
Ulam spiral of MultiplicativeOrder :
Tech Notes
Related Guides
Related Links
History
Introduced in 1999 (4.0) | Updated in 2023 (13.3)
Text
Wolfram Research (1999), MultiplicativeOrder, Wolfram Language function, https://reference.wolfram.com/language/ref/MultiplicativeOrder.html (updated 2023).
CMS
Wolfram Language. 1999. "MultiplicativeOrder." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/MultiplicativeOrder.html.
APA
Wolfram Language. (1999). MultiplicativeOrder. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MultiplicativeOrder.html
BibTeX
@misc{reference.wolfram_2025_multiplicativeorder, author="Wolfram Research", title="{MultiplicativeOrder}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/MultiplicativeOrder.html}", note=[Accessed: 12-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_multiplicativeorder, organization={Wolfram Research}, title={MultiplicativeOrder}, year={2023}, url={https://reference.wolfram.com/language/ref/MultiplicativeOrder.html}, note=[Accessed: 12-November-2025]}