Tool to compute the modular inverse of a number. The modular multiplicative inverse of an integer N modulo m is an integer n such as the inverse of N modulo m equals n.
Modular Multiplicative Inverse - dCode
Tag(s) : Arithmetics
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Modular Multiplicative Inverse
Modular Inverse Calculator
Batch InvMod Calculator
Answers to Questions (FAQ)
What is the modular Inverse? (Definition)
The value of the modular inverse of $ a $ by the modulo $ n $ is the value $ a^{-1} $ such that $ a \cdot a^{-1} \equiv 1 \pmod n $
It is common to note this modular inverse $ u $ and to use these equations $$ u \equiv a^{-1} \pmod n \\ a u \equiv 1 \pmod n $$
If a modular inverse exists then it is unique.
How to calculate a modular inverse?
To calculate the value of the modulo inverse, use the extended euclidean algorithm which finds solutions to the Bezout identity $ au + bv = \text{G.C.D.}(a, b) $. Here, the gcd value is known, it is 1: $ \text{G.C.D.}(a, b) = 1 $, thus, only the value of $ u $ is needed.
Example: $ 3^{-1} \equiv 4 \mod 11 $ because $ 4 \times 3 = 12 $ and $ 12 \equiv 1 \mod 11 $
dCode uses the Extended Euclidean algorithm for its inverse modulo N calculator and arbitrary precision functions to get results with big integers.
How to calculate v in au+bv?
Use the Bezout identity, also available on dCode.
What does invmod mean?
The keyword invmod is the abbreviation of inverse modular.
What is a multiplicative inverse?
A multiplicative inverse is the other name of a modular inverse.
Source code
dCode retains ownership of the "Modular Multiplicative Inverse" source code. Any algorithm for the "Modular Multiplicative Inverse" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Modular Multiplicative Inverse" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Modular Multiplicative Inverse" or any other element are not public (except explicit open source licence). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Modular Multiplicative Inverse" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source (Creative Commons CC-BY free distribution license).
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Modular Multiplicative Inverse on dCode.fr [online website], retrieved on 2025-07-15,
