Wikipedia
en.wikipedia.org › wiki › Modulo
Modulo - Wikipedia
6 days ago - In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division).
GeeksforGeeks
geeksforgeeks.org › dsa › modular-division
Modular Division - GeeksforGeeks
July 14, 2025 - Modular division is defined when the modular inverse of the divisor exists. The inverse of an integer 'x' is another integer 'y' such that (x*y) % m = 1 where m is the modulus. When does inverse exist?
elementary number theory - Help me understand division in modular arithmetic - Mathematics Stack Exchange
Now from what i understand This is perfectly fine (48/8)mod 8=6 right ! but as it says in above so when we modular division is not valid (a/b) modulo n when b is multiple of n. More on math.stackexchange.com
algorithm - Modulo of Division of Two Numbers - Stack Overflow
we know that (A + B) % P = (A % P + B % P) % P (A * B) % P = (A % P * B % P) % P where P is a prime . I need to calculate (A / B) % P where A,B can be very large and can overflow . Does such kin... More on stackoverflow.com
What's the difference between % (modulo) and // (floor division)
There's a function called divmod that does both, and if you use that a few times, you'll get a feel for the difference. For example, let's say you have a quantity of money, and you want to make change. cents = 93 quarters, cents = divmod(cents, 25) dimes, cents = divmod(cents, 10) nickels, cents = divmod(cents, 5) At each step, we take as many chunks as we can (that's the div), then we keep any leftovers (that's the mod) More on reddit.com
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September 15, 2024Modulus Operator
Hi Everyone, I have been doing some self learning for C++. I cant seem to wrap my head around the logic of using the modulus operator to test whether an integer parameter is odd or even. The code i have got is #includ… More on forum.arduino.cc
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August 5, 2021Videos
08:56
Modulo Division of polynomials - YouTube
08:05
Division of Modular Arithmetic|Modular Arithmetic Equations - YouTube
04:42
Modular Arithmetic: A New Way To Divide - YouTube
06:20
Python Tutorial #5: Python Division: /, //, and % 🐍 - YouTube
07:42
Division and Modulo Operators - Intro to Programming for Kids and ...
18:39
Basics of Modular Arithmetic - YouTube
CalculatorSoup
calculatorsoup.com › calculators › math › modulo-calculator.php
Modulo Calculator
Modulo calculator finds a mod b, the remainder when a is divided by b. The modulo operation returns the remainder in division of 2 positive or negative numbers or decimals.
Top answer 1 of 2
4
There is a simple definition of the 
operator, in that if we have 
, then we may write 
for some 
.
We reduce 
to a value in 
and fix 
accordingly.
So:
for some 
.
And:
$$a \pmod n + b \pmod n = a+k_an + b+k_bn = (a+b) + (k_a+k_b)n = (a+b) + kn$$
So these two are the same.
With division, the two resulting formula are not the same:
$$\frac{a \pmod n}{b \pmod n} = \frac{a + k_an}{b + k_bn}$$
2 of 2
2
I'll try my best to answer concisely:
It is not a condition for
to be in 
but since every integer is equivalent to one, it makes sense to use numbers in this rangeWhen I studied this, I liked to think that rather than division of
by 
, we multiply by the inverse of 
. If 
there is no inverse (it is 'like' dividing by zero in a naïve sense)
To find the inverse of 
there is the extended Euclidean algorithm.
Top answer 1 of 4
59
Yes, but it's different:
(a - b) mod p = ((a mod p - b mod p) + p) mod p
(a / b) mod p = ((a mod p) * (b^(-1) mod p)) mod p
Where b^(-1) mod p is the modular inverse of b mod p. For p = prime, b^(-1) mod p = b^(p - 2) mod p.
Edit:
(N*(N^2+5)/6)%P
You don't need any modular inverses from this. Just simplify the fraction: N or N^2+5 will be divisible by 2 and 3. So divide them and then you have (a*b) mod P.
2 of 4
7
Vlad's answer is correct:
(a - b) mod p = ((a mod p - b mod p) + p) mod p
(a / b) mod p = ((a mod p) * (b^(-1) mod p)) mod p
These and some other operations are outlined here in the Equivalencies section.
Just want to let you know that this will work not only for prime number p. The first one will work for any p. The second one will work for any p, where b^(-1) or modular inverse is defined.
The modular inverse can be calculated with extended Euclidian algorithm
Khan Academy
khanacademy.org › computing › computer-science › cryptography › modarithmetic › a › what-is-modular-arithmetic
Khan Academy
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University of Vermont
uvm.edu › ~cbcafier › cs1210 › book › 04_variables,_statements,_and_expressions › modulo_and_floor.html
Modulo, floor division, and modular arithmetic – Clayton Cafiero
June 26, 2025 - When presenting expressions, we saw examples of common arithmetic operations: addition, subtraction, multiplication, and division. Here we will present two additional operations which are closely related: the modulo or “remainder” operator, and what’s variously called “quotient”, “floor division”, “integer division” or “Euclidean division.”1
Reddit
reddit.com › r/learnpython › what's the difference between % (modulo) and // (floor division)
r/learnpython on Reddit: What's the difference between % (modulo) and // (floor division)
September 15, 2024 -
Hello, Can someone please explain the me the difference between those two arithmetic signs because am new to programming python:)
Top answer 1 of 5
8
There's a function called divmod that does both, and if you use that a few times, you'll get a feel for the difference. For example, let's say you have a quantity of money, and you want to make change. cents = 93 quarters, cents = divmod(cents, 25) dimes, cents = divmod(cents, 10) nickels, cents = divmod(cents, 5) At each step, we take as many chunks as we can (that's the div), then we keep any leftovers (that's the mod)
2 of 5
6
101 // 10 = 10 101 % 10 = 1 The floor division is like "how often does the 10 fit into the 101". Or you can think about it as a regular division in combination with cutting off the decimal point. Modulo on the other hand is "how often does the 10 fit into the 101 and give me what is left". The modulo operator returns the remainder of the division.
A Security Site
asecuritysite.com › comms › mod_div
Binary division (modulo 2)
Bib: @misc{asecuritysite_43288, title = {Binary division (modulo 2)}, year={2026}, organization = {Asecuritysite.com}, author = {Buchanan, William J}, url = {https://asecuritysite.com/comms/mod_div}, note={Accessed: March 06, 2026}, howpublished={\url{https://asecuritysite.com/comms/mod_div}} }
MathWorks
mathworks.com › matlab › mathematics › elementary math › arithmetic operations
mod - Remainder after division (modulo operation) - MATLAB
b = mod(a,m) returns the remainder after division of a by m, where a is the dividend and m is the divisor. This function is often called the modulo operation, which can be expressed as b = a - m.*floor(a./m).
PHP
php.net › manual › en › language.operators.arithmetic.php
PHP: Arithmetic - Manual
The division operator / returns a float value unless the two operands are int (or numeric strings which are type juggled to int) and the numerator is a multiple of the divisor, in which case an integer value will be returned. For integer division, see intdiv(). Operands of modulo are converted to int before processing.
Stanford Crypto
crypto.stanford.edu › pbc › notes › numbertheory › arith.html
Number Theory - Modular Arithmetic
Which answer should we choose for "\(4 / 2\)", \(5\) or \(2\)? We could introduce some arbitrary convention, such as choosing the smallest answer when considering the least residue as an integer, but then division will behave strangely. Instead, we require uniqueness, that is \(x\) divided by \(y\) modulo \(n\) is only defined when there is a unique \(z \in \mathbb{Z}_n\) such that \(x = y z\).
The Teclado Blog
blog.teclado.com › pythons-modulo-operator-and-floor-division
Python's modulo operator and floor division
October 26, 2022 - We care about the whole number result of the division (in this case 3), which is called the quotient. The amount left over after the division is called the remainder, which in this case is 1. To give you another example, 11 / 3 gives us a quotient of 3 and a remainder of 2. In Python, the modulo operator simply yields the remainder:
Wolfram Language
reference.wolfram.com › language › ref › Mod.html
Mod: Get the remainder on division—Wolfram Documentation
Mod[m, n] gives the remainder on division of m by n. Mod[m, n, d] uses an offset d.
GeeksforGeeks
geeksforgeeks.org › python › what-is-a-modulo-operator-in-python
Modulo operator (%) in Python - GeeksforGeeks
December 20, 2025 - Here, a is divided by b, and the modulo operator returns the remainder left after the division.
Quora
quora.com › What-is-the-difference-between-using-a-modulo-operator-and-division-by-its-inverse-in-programming-languages-Is-there-a-practical-reason-for-choosing-one-over-the-other
What is the difference between using a modulo operator and division by its inverse in programming languages? Is there a practical reason for choosing one over the other? - Quora
Answer (1 of 2): The problem with division (and modulo) is that they are often one of the slowest operations that can be performed on a processor. If you’re working with processors for embedded systems, there’s sometimes hardware for making multiplication in just over the same time it takes to ma...
Arduino Forum
forum.arduino.cc › projects › programming
Modulus Operator - Programming - Arduino Forum
August 5, 2021 - Hi Everyone, I have been doing some self learning for C++. I cant seem to wrap my head around the logic of using the modulus operator to test whether an integer parameter is odd or even. The code i have got is #include #include bool isEven(int x) { return (x % 2) == 0; } int main() { std::cout
Real Python
realpython.com › python-modulo-operator
Python Modulo Operator (%): How to Use Mod in Python – Real Python
April 1, 2023 - In this section, you’ll see how you can use the modulo operator to determine if a number is even or odd. Using the modulo operator with a modulus of 2, you can check any number to see if it’s evenly divisible by 2.
Codeforces
codeforces.com › blog › entry › 72527
Modular Arithmetic for Beginners - Codeforces
Again, you generally never need to store the fractions as their "true" values, only their residues modulo $$$m$$$. Congratulations! You have now mastered $$$\mathbb Z / p \mathbb Z$$$ field arithmetic! A "field" is just a fancy term from abstract algebra theory for a set with the four basic operators (addition, subtraction, multiplication, division) defined in a way that works just like you've learned in high-school for the rational and real numbers (however division by zero is still undefined), and $$$\mathbb Z / p \mathbb Z$$$ is just a fancy term meaning the set of integers from $$$0$$$ to $$$p - 1$$$ treated as residues modulo $$$p$$$.
Omni Calculator
omnicalculator.com › math › modulo
Modulo Calculator
May 8, 2025 - Multiply the divisor by the quotient. So it's 10 × 24 = 240 in our example. Subtract this number from your initial number (dividend). Here: 250 - 240 = 10. The number you obtain is the result of the modulo operation.