Mod
Details
- Mod is also known as modulo operation.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- Typically used in modular arithmetic, cryptography, random number generation and cyclic operations in programs.
- Mod [m,n] gives the remainder of m divided by n.
- Mod [m,n] is equivalent to m-nQuotient [m,n].
- For positive integers m and n, Mod [m,n] is an integer between 0 and n-1.
- Mod [m,n,d] gives a result such that and .
Examples
open all close allBasic Examples (4)
Compute 5 mod 3:
The remainder on division of 5 by 3 offset to start with 1:
Plot the sequence with fixed modulus:
Plot the sequence, varying the modulus:
Scope (13)
Numerical Evaluation (6)
Compute using integers:
Compute using an offset:
Mod works on integers:
Rational numbers:
Real numbers:
Complex numbers:
Exact numbers:
Inexact number:
Compute using large integers:
Mod threads over lists:
TraditionalForm formatting:
Applications (19)
Basic Applications (3)
Numeric Identifiers (1)
Given an International Standard Book Number (ISBN), check whether or not it is valid:
An ISBN is valid if , where each is the digit of the ISBN:
Check if each of the ISBNs are valid:
Cryptography (2)
Build an RSA-like encryption scheme. Start with the modulus:
Find the universal exponent of the multiplication group modulo n:
Private key:
Public key:
Encrypt a message:
Decrypt it:
Use Mod to create a Caesar cipher that shifts letters in the alphabet to encrypt a message:
Encrypt a message with a key:
Decrypt the message:
Number Theory (6)
Check if numbers of the form are prime or composite:
Select primes below 100 having the form of :
Fermat's little theorem:
Euler's theorem:
Wilson's theorem:
Define a notation for addition modulo 2:
Use Mod to solve systems of linear congruences:
Computer Sciences (3)
Create a random number generator that uses the current time as a seed:
Choose a modulus and base:
Compute 1000 random numbers between 0 and 1:
Extract parts of a list cyclically:
Modular computation of a matrix inverse:
First compute the matrix adjoint:
Then compute the modular inverse of a matrix:
Check that the inverse gives the correct result:
Politics, Economics and Social Sciences (2)
Assign memory addresses to social security numbers based on a hashing algorithm:
Assign each social a location, ensuring that there are no collisions:
Compute the result:
Compute the hash of a single social security number:
Other Applications (2)
Simulate a particle bouncing in a noncommensurate box:
System of 12-tone equal temperament:
Notes that have a difference of 1200 cents are considered to be from the same congruence class:
Properties & Relations (7)
Mod is a periodic function:
Mod is defined over all complex numbers:
Range:
Mod is transitive. If and , then :
If divides then :
The QuotientRemainder [a,n] is the same as Mod [a,n]:
Use PowerMod to compute the modular inverse:
Check the result:
The results have the same sign as the modulus:
For a positive real number x, Mod [x,1] gives the fractional part of x:
Possible Issues (1)
Some computations may require higher internal precision than the default:
Reset the value of $MaxExtraPrecision :
Neat Examples (4)
Binomial coefficients modulo 2:
Additive cellular automaton:
Plot of an Ulam spiral where numbers are colored based on their congruence modulo 49:
Modular addition tables:
History
Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 1999 (4.0) ▪ 2000 (4.1) ▪ 2002 (4.2)
Text
Wolfram Research (1988), Mod, Wolfram Language function, https://reference.wolfram.com/language/ref/Mod.html (updated 2002).
CMS
Wolfram Language. 1988. "Mod." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2002. https://reference.wolfram.com/language/ref/Mod.html.
APA
Wolfram Language. (1988). Mod. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Mod.html
BibTeX
@misc{reference.wolfram_2025_mod, author="Wolfram Research", title="{Mod}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/Mod.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_mod, organization={Wolfram Research}, title={Mod}, year={2002}, url={https://reference.wolfram.com/language/ref/Mod.html}, note=[Accessed: 17-November-2025]}