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Computational operation
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the
Modulo
Computation modulo a fixed integer
book Disquisitiones Arithmeticae, published in 1801. Modular arithmetic modulo m consists of systematically replacing the results of additions, multiplications
Modular_arithmetic
Módulo is a Brazilian company with international operations specializing in technology for Governance, Risk and Compliance. It operates in areas of software
Módulo
Word with multiple distinct meanings
In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert
Modulo_(mathematics)
Japanese manga series
Jujutsu Kaisen Modulo (Japanese: 呪術廻戦≡（モジュロ）, Hepburn: Jujutsu Kaisen Mojuro) is a Japanese manga series written by Gege Akutami and illustrated by Yūji
Jujutsu_Kaisen_Modulo
Italian concept sports car
S Modulo is a concept sports car designed by Paolo Martin of the Italian carrozzeria Pininfarina, unveiled at the 1970 Geneva Motor Show. The Modulo has
Ferrari_Modulo
Modular arithmetic concept
primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. In symbols, g is a primitive root modulo n if for every
Primitive_root_modulo_n
Topics referred to by the same term
Look up modulo in Wiktionary, the free dictionary. Modulo is the remainder operation, which carries out a division operation with a remainder as result
Modulo_(disambiguation)
Integer that is a perfect square modulo some integer
number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that
Quadratic_residue
Group of units of the ring of integers modulo n
non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the
Root_of_unity_modulo_n
Unique numeric book identifier since 1970
(11 minus the remainder of the sum of the products modulo 11) modulo 11. Taking the remainder modulo 11 a second time accounts for the possibility that
ISBN
Concept in modular arithmetic
remainder after dividing ax by the integer m is 1. If a does have an inverse modulo m, then there is an infinite number of solutions of this congruence, which
Modular multiplicative inverse
Modular_multiplicative_inverse
Theorem on polynomial roots modulo prime powers
univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. (The lifting
Hensel's_lemma
Topics referred to by the same term
Modulus (digital counter), the number of states in a counter's count sequence Modulo operation (a % b, mod(a, b), etc.), in both math and programming languages;
Modulus
Logical problem studied in computer science
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable
Satisfiability modulo theories
Satisfiability_modulo_theories
Natural number
reduction modulo 2 records the parity of an integer: even integers are congruent to 0 modulo 2, and odd integers are congruent to 1 modulo 2. In algebra
2
Topics referred to by the same term
partitioning attack applicable to block and stream ciphers Modulo (mathematics) Modular arithmetic Modulo operation Modular exponentiation MOD., a science museum
Mod
Period of the Fibonacci sequence modulo an integer
π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known
Pisano_period
Number system extending the rational numbers
arithmetic modulo a positive integer n consists of "approximating" every integer by the remainder of its division by n, called its residue modulo n. The main
P-adic_number
Mathematical concept
general, may not be reduced modulo 2). It is then possible to reduce all coefficients modulo 2, which will give a modular form modulo 2. Modular forms are generated
Modular_forms_modulo_p
Algebraic structure
prime field of order p {\displaystyle p} may be constructed as the integers modulo p {\displaystyle p} , Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} }
Finite_field
Type of number sequence
a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted
Equidistributed_sequence
Set of residue classes modulo n, relatively prime to n
reduced residue system modulo n if: gcd(r, n) = 1 for each r in R, R contains φ(n) elements, no two elements of R are congruent modulo n. Here φ denotes Euler's
Reduced_residue_system
Error detection for identification numbers
check digit method would be to take the sum of all digits (digital sum) modulo 10. This would catch any single-digit error, as such an error would always
Check_digit
Japanese manga series
and illustrated by manga artist Yūji Iwasaki [ja], titled Jujutsu Kaisen Modulo, ran in Weekly Shōnen Jump from September 2025 to March 2026. By December
Jujutsu_Kaisen
Football championship of Minas Gerais, Brazil
The 2026 Campeonato Mineiro (officially Campeonato Mineiro SICOOB 2026 – Módulo I for sponsorship reasons) was the 112th edition of the state championship
2026_Campeonato_Mineiro
Lossy Compression Algorithm
Modulo-N code is a lossy compression algorithm used to compress correlated data sources using modular arithmetic. When applied to two nodes in a network
Modulo-N_code
Concept in modular arithmetic
multiplicative order of a modulo n is the smallest positive integer k such that ak ≡ 1 (mod n). In other words, the multiplicative order of a modulo n is the order
Multiplicative_order
Minivan produced by Honda
(facelift) 2007–2009 Honda Stepwgn Spada (facelift) The Modulo Stepwgn Concept and Stepwgn Modulo Concept X Final Room are concept cars based on the Stepwgn
Honda_Stepwgn
Computer arithmetic error
the resulting bit-pattern is the same as if the operation was performed modulo 2W, where W is the word size in bits. The operation also sets or unsets
Integer_overflow
Problem of inverting exponentiation in groups
that b k = a {\displaystyle b^{k}=a} . In the special case of arithmetic modulo an integer m {\displaystyle m} , the more commonly used term is index: One
Discrete_logarithm
Motor vehicle
S660 Concept Edition Honda S660 α Honda S660 Modulo Honda S660 Modulo Honda S660 Modulo X Honda S660 Modulo X Interior S07A Turbo engine Takahashi, Yoshio
Honda_S660
Group obtained by aggregating similar elements of a larger group
structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying
Quotient_group
Function in mathematical number theory
algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n. As this is a finite abelian group, there must exist an element whose
Carmichael_function
Function in number theory
= { 1 if a is a quadratic residue modulo p and a ≢ 0 ( mod p ) , − 1 if a is a quadratic nonresidue modulo p , 0 if a ≡ 0 ( mod p ) . {\displaystyle
Legendre_symbol
The mining project of Aclara Resources, known as Penco Module (Spanish: Módulo Penco), had as of March 2026 its environmental impact assessment under evaluation
Rare-earth_resources_in_Chile
Football championship of Minas Gerais, Brazil
The 2025 Campeonato Mineiro (officially Campeonato Mineiro SICOOB 2025 – Módulo I for sponsorship reasons) was the 111th edition of the state championship
2025_Campeonato_Mineiro
Football club in Minas Gerais, Brazil
better regular season. In August 2025, North won the Campeonato Mineiro Módulo II with a single goal by Luiz Thiago in the two-legged final against URT
North_Esporte_Clube
Subroutine call performed as final action of a procedure
argument (product in the above example) to the function. Tail recursion modulo cons is a generalization of tail-recursion optimization introduced by David
Tail_call
In mathematics, invertible homomorphism
element is an integer modulo 2 and the second element is an integer modulo 3, with component-wise addition and multiplication modulo 2 and 3. These rings
Isomorphism
Mathematical statement of uniqueness, except for an equivalent structure
informal contexts, mathematicians often use the word modulo (or simply mod) for similar purposes, as in "modulo isomorphism". Objects that are distinct up to
Up_to
Theorem on modular exponentiation
n ) {\displaystyle a^{\varphi (n)}} is congruent to 1 {\displaystyle 1} modulo n, where φ {\displaystyle \varphi } denotes Euler's totient function; that
Euler's_theorem
Topics referred to by the same term
'+' denotes 'exclusive or' operation, or in a quotient ring of numbers modulo 2) The terms 1+1, One Plus One, or One and One may refer to: 1 + 1 + 1 +
1+1
List of security identifiers
The check digit is then calculated by: [10 − (126 modulo 10)] modulo 10 = (10 − 6) modulo 10 = 4 modulo 10 = 4 "SEDOL Masterfile - Service & Technical Guide"
SEDOL
Arithmetic in a field with a finite number of elements
Galois. GF(p), where p is a prime number, is simply the ring of integers modulo p. That is, one can perform operations (addition, subtraction, multiplication)
Finite_field_arithmetic
Type of linear congruential generator with no additive constant
congruential generator (LCG) that operates in multiplicative group of integers modulo n. The general formula is X k + 1 = a ⋅ X k mod m , {\displaystyle X_{k+1}=a\cdot
Lehmer random number generator
Lehmer_random_number_generator
28-year cycle of the Julian calendar
also be written as: (year number + 8) modulo 28) + 1. The position of 2026 in the solar cycle is ((2026 + 8) modulo 28) + 1 = 18 + 1 = 19. This cycle also
Solar_cycle_(calendar)
Mathematical sequence
multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1. In other words, the sequence of the fractional parts of each term will
Weyl_sequence
Substitution cipher based on linear algebra
matrices (modulo 26). The cipher can, of course, be adapted to an alphabet with any number of letters; all arithmetic just needs to be done modulo the number
Hill_cipher
Irreducible polynomial whose roots are nth roots of unity
multiplicative order of p modulo n. In particular, Φ n {\displaystyle \Phi _{n}} is irreducible if and only if p is a primitive root modulo n, that is, p does
Cyclotomic_polynomial
Formula concerning prime numbers
is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to
Euler's_criterion
State football league of Minas Gerais, Brazil
and Tostão had their professional football debut in the competition. 2025 Módulo I América (Belo Horizonte) Athletic (São João del-Rei) Atlético (Belo Horizonte)
Campeonato_Mineiro
How many times a number is divisible by 2
ν2(n) or ord2(n). It is not to be confused with the multiplicative order modulo 2. The 2-order provides a unified description of various classes of integers
Singly_and_doubly_even
Factorization method based on the difference of two squares
all the values for a. Squares are always congruent to 0, 1, 4, 5, 9, 16 modulo 20, because these are the quadratic residues of 20. The values repeat with
Fermat's_factorization_method
Gives conditions for the solvability of quadratic equations modulo prime numbers
arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most
Quadratic_reciprocity
Brazilian association football club based in Betim, Minas Gerais, Brazil
the Campeonato Mineiro Módulo II. In 2020, Betim narrowly missed out a consecutive promotion, finishing third in the year's Módulo II. After a fifth place
Betim_Futebol
Software for solving satisfiability problems
Z3, also known as the Z3 Theorem Prover, is a satisfiability modulo theories (SMT) solver developed by Microsoft. Z3 was developed in the Research in Software
Z3_Theorem_Prover
Problem in number theory
Unsolved problem in mathematics Is there a number that is not 4 or 5 modulo 9 and that cannot be expressed as a sum of three cubes? More unsolved problems
Sums_of_three_cubes
Algorithm for fast modular multiplication
RSA and Diffie–Hellman key exchange are based on arithmetic operations modulo a large odd number, and for these cryptosystems, computations using Montgomery
Montgomery modular multiplication
Montgomery_modular_multiplication
Soccer club
2016, they competed in the Campeonato Mineiro — Módulo I, however, in 2017 they were relegated to Modulo II. "CLUBE ATLÉTICO TRICORDIANO". FMF. Archived
Clube_Atlético_Tricordiano
Compiler optimization technique
Global scheduling: instructions can move across basic block boundaries. Modulo scheduling: an algorithm for generating software pipelining, which is a
Instruction_scheduling
Mapping arbitrary data to fixed-size values
+ r0 is any nonzero polynomial modulo 2 with at most t nonzero coefficients, then R(x) is not a multiple of P(x) modulo 2. If follows that the corresponding
Hash_function
Algorithm for generating pseudo-randomized numbers
the parameters m and a. For example, a = 1 and c = 1 produces a simple modulo-m counter, which has a long period, but is obviously non-random. Other values
Linear_congruential_generator
Esoteric, minimalist programming language
unsigned integer (0–255) and is initialized to zero. Arithmetic on cells wraps modulo 256, so incrementing 255 produces 0 and decrementing 0 produces 255. A data
Brainfuck
Algorithm used in modular arithmetic
find a square root of n modulo p. The Tonelli–Shanks algorithm cannot be used for composite moduli: finding square roots modulo composite numbers is a
Tonelli–Shanks_algorithm
Type of generating function in mathematics
of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on. For a prime number p let K be a p-adic field, i.e
Igusa_zeta_function
defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the
Algebraic-group factorization algorithm
Algebraic-group_factorization_algorithm
Algorithm for shuffling a finite sequence
cost of eliminating "modulo bias" when generating random integers for a Fisher–Yates shuffle depends on the approach (classic modulo, floating-point multiplication
Fisher–Yates_shuffle
Italian car designer widely (born 1943)
Tomaso/Ghia where he styled the Ferrari Dino Berlinetta Competizione, Ferrari Modulo concept, Fiat 130 Coupé and the Rolls-Royce Camargue. Born in Turin, Italy
Paolo_Martin
Methods of error detection and correction in communications
division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero
Mathematics of cyclic redundancy checks
Mathematics_of_cyclic_redundancy_checks
State football league of Minas Gerais, Brazil
The Campeonato Mineiro Módulo II is the second tier of the professional state football league in the Brazilian state of Minas Gerais. It is run by the
Campeonato_Mineiro_Módulo_II
Country in South America
Abandonados 70% de módulos de BA Archived 27 September 2007 at the Wayback Machine Diario 2001 (29 July 2007). "El 80% de los módulos de Barrio Adentro
Venezuela
Theorem on prime numbers
{\displaystyle -1} modulo n {\displaystyle {n}} . Then ( n − 1 ) ! {\displaystyle (n-1)!} would also be congruent to − 1 {\displaystyle -1} modulo q {\displaystyle
Wilson's_theorem
reduction Fundamental to local analysis in arithmetic problems is to reduce modulo all prime numbers p or, more generally, prime ideals. In the typical situation
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
(modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its
Fundamental unit (number theory)
Fundamental_unit_(number_theory)
Computer checksum algorithm
beginning of an Adler-32 run, A is initialized to 1, B to 0. The sums are done modulo 65521 (the largest prime number smaller than 216). The bytes are stored
Adler-32
American domestic terrorist (1942–2023)
four reading phases (horizontal, vertical, diagonal). Decryption required modulo 90 addition of ciphertext numbers to the key, followed by substitution using
Ted_Kaczynski
Sporadic simple group
In the area of modern algebra known as group theory, the Janko group J1 is a sporadic simple group of order 175 , 560 = 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19 ≈ 2 ×
Janko_group_J1
Generalization of the Legendre symbol in number theory
residue modulo n. If a is a quadratic nonresidue modulo n, then (a/n) may be −1 or +1. This is because for a to be a quadratic residue modulo n, it has
Jacobi_symbol
Algorithm for generating prime numbers
L. Atkin and Daniel J. Bernstein. In the algorithm: All remainders are modulo-sixty remainders (divide the number by 60 and return the remainder). All
Sieve_of_Atkin
Conjecture in number theory
given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density
Artin's conjecture on primitive roots
Artin's_conjecture_on_primitive_roots
General-purpose programming language
operators (+, -, *, /), the floor-division operator //, and the modulo operator %. (With the modulo operator, a remainder can be negative, e.g., 4 % -3 == -2
Python_(programming_language)
Attack applicable to block and stream ciphers
in how the cipher operates over equivalence classes (congruence classes) modulo n. The method was first suggested in 1999 by John Kelsey, Bruce Schneier
Mod_n_cryptanalysis
Probabilistic primality test
a prime, then the only square roots of 1 modulo n are 1 and −1. Proof Certainly 1 and −1, when squared modulo n, always yield 1. It remains to show that
Miller–Rabin_primality_test
Algorithm for public-key cryptography
formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way
RSA_cryptosystem
Special-purpose algorithm for factoring integers
working in the multiplicative group modulo a composite number N, we are also working in the multiplicative groups modulo all of N's factors. The existence
Pollard's_p_−_1_algorithm
Algorithm for determining whether a number is prime
calculate an − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime. If an−1 (modulo n) is 1 but n is not
Primality_test
Repeatable pattern of differences between prime numbers
exist a prime p such that the tuple includes every different possible value modulo p. If such a prime p existed, then no matter which value of n was chosen
Prime_k-tuple
Mathematical plot of the Goldbach function
Golbach's comet up to the integer 2000. Red points are multiples of 6, blue points are integers that are 2 modulo 6, and green points are 4 modulo 6.
Goldbach's_comet
A prime p divides a^p–a for any integer a
odd prime, then the integers modulo p form a finite field, in which 1 modulo p has exactly two square roots, 1 and −1 modulo p. Note that ad ≡ 1 (mod p)
Fermat's_little_theorem
Brazilian association football club based in São João del-Rei, Minas Gerais, Brazil
eliminated Valeriodoce, and gained access to the 2019 Campeonato Mineiro Módulo II. Athletic ended in second place, after losing the final on penalties
Athletic_Club_(MG)
Japanese manga artist (born 1992)
Shōnen Jump Shueisha (Japanese) Viz Media (English) Manga Jujutsu Kaisen Modulo (呪術廻戦≡（モジュロ）, Jujutsu Kaisen Mojuro) 2025–2026 Weekly Shōnen Jump Shueisha
Gege_Akutami
Even integers as sums of two primes
that the number of these representations depends strongly on the value modulo 3 of the number. Although Goldbach's conjecture implies that every positive
Goldbach's_conjecture
size (modulo 128 and modulo 32768) where the maximum number of outstanding frames for acknowledgment is raised from 7 (modulo 8) to 127 (modulo 128) and
LAPB
Complex number whose real and imaginary parts are both integers
{\displaystyle k} is odd (in particular, a norm is not itself congruent to 3 modulo 4). The norm is multiplicative, that is, one has N ( z w ) = N ( z ) N (
Gaussian_integer
using refinement types. Properties are verified using a satisfiability modulo theories (SMT) solver which is SMTLIB2-compliant, such as the Z3 Theorem
Liquid_Haskell
Amount left over after computation
the polynomial "left over" after dividing one polynomial by another. The modulo operation is the operation that produces such a remainder when given a dividend
Remainder
Unsolved problem in cryptography
only the public key. The RSA algorithm raises a message to an exponent, modulo a composite number N whose factors are not known. Thus, the task can be
RSA_problem
Quantities used in astronomy
one calculates the mean anomaly M(t) at a time t: M = (t − T) · 360°/P (modulo 360°) Next, find the eccentric anomaly E, such that E = M + e sin(E) (Kepler's
Thiele−Innes_elements
travel, tourism, insurance
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travel, tourism, insurance