Khan Academy
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Congruence modulo (article) | Cryptography
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system of algebraic operations defined for remainders under division by a fixed positive integer; system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus
Wikipedia
en.wikipedia.org › wiki › Modular_arithmetic
Modular arithmetic - Wikipedia
2 weeks ago - If a ≡ b (mod m) and a−1 exists, then a−1 ≡ b−1 (mod m) (compatibility with multiplicative inverse, and, if a = b, uniqueness modulo m). If ax ≡ b (mod m) and a is coprime to m, then the solution to this linear congruence is given by x ≡ a−1b (mod m).
math mode - How to write the congruence modulo n symbol? - TeX - LaTeX Stack Exchange
When a is the remainder when b is divided by n, we say b is congruent to a mod n. How is it typeset using LaTeX? More on tex.stackexchange.com
Confusion regarding the symbol '≡' (congruent to) in modular arithmetic
In this context neither "≡" or "mod" is a binary operator that produces a result, like you might be used to seeing in programming languages or other parts of mathematics. xe ≡ y mod m just means that the remainder of xe divided by m is the same as the remainder of y divided by m ("xe is congruent to y modulo m"). You can switch the items on each side of the ≡ (before the "mod m") without changing the meaning of the statement. More on reddit.com
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July 15, 2025notation - Why do we use "congruent to" instead of equal to? - Mathematics Stack Exchange
For example, ElGamal signature ... in the first formula, $r$ really needs to be chosen as the least non-negative representative of its congruence class modulo $p$, while the second formula involves a modular inverse modulo $p-1$. $\endgroup$... More on math.stackexchange.com
modular arithmetic - Notation for modulo: congruence relation vs operator - Mathematics Stack Exchange
If a and b are congruent modulo a number c, we might write $a \equiv b \pmod c$. When writing programs, it's often useful to compute the remainder after division, and in pseudocode we might write a... More on math.stackexchange.com
Videos
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Number Theory | Congruence Modulo n -- Definition and Examples ...
Definition of Congruence Modulo m
08:18
What is Modular Congruence? | Congruence Modulo n, Modular Congruence ...
Introduction to Congruence Modulo n
21:06
Proving Properties of Modular Congruence - YouTube
18:13
Modular Arithmetic and Modulo Congruence - YouTube
Wolfram MathWorld
mathworld.wolfram.com › Congruence.html
Congruence -- from Wolfram MathWorld
June 6, 2024 - If two numbers b and c have the ... and the statement "b is congruent to c (modulo m)" is written mathematically as b=c (mod m). (1) If b-c is not integrally divisible by m, then it is said that "b is not congruent to ...
Top answer 1 of 2
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There are several ways, choose the one you prefer.
\documentclass{article}
\usepackage{amsmath}
\renewcommand{\arraystretch}{1.5} % just to make the lines spread out
\begin{document}
\begin{tabular}{lll}
\verb|
| & 
\\
\verb|
| & 
\\
\verb|
| & 
\\
\verb|
| & 
& (wrong)
\end{tabular}
\end{document}
Don't forget the braces: try a\equiv b \pmod 11 or a\equiv b \pmod pq and see why.
The last one is marked “wrong”, because the usage is improper: \bmod should be used for the “modulo” binary operation (the one that is often denoted by % in computing).
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Use pmod n. It is known to work.
University of Washington
sites.math.washington.edu › ~greenber › Congruences.pdf pdf
BASIC PROPERTIES OF CONGRUENCES
BASIC PROPERTIES OF CONGRUENCES · The letters a, b, c, d, k represent integers. The letters m, n represent positive integers. The · notation a ≡b (mod m) means that m divides a −b. We then say that a is congruent to b · modulo m. 1. (Reflexive Property): a ≡a (mod m) 2.
Reddit
reddit.com › r/cryptography › confusion regarding the symbol '≡' (congruent to) in modular arithmetic
r/cryptography on Reddit: Confusion regarding the symbol '≡' (congruent to) in modular arithmetic
July 15, 2025 -
Hello everyone,
In modular arithmetic, if we know the remainder r when dividing a by m, we write it as:
a ≡ r mod m
As I understand it, r is the result of the operation a mod m.
However, in other formulas—like in RSA encryption—we often see something like:
y ≡ x^(e) mod m
This means that y is the result of the operation x^(e) mod n.
So to me, it would feel more intuitive to write:
x^(e) ≡ y mod n
since x^(e) mod n = y, and the expression being reduced appears on the left-hand side.
The way the modular expression is written can be a little confusing at first, but both forms describe the same relationship.
Top answer 1 of 5
8
In this context neither "≡" or "mod" is a binary operator that produces a result, like you might be used to seeing in programming languages or other parts of mathematics. xe ≡ y mod m just means that the remainder of xe divided by m is the same as the remainder of y divided by m ("xe is congruent to y modulo m"). You can switch the items on each side of the ≡ (before the "mod m") without changing the meaning of the statement.
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This means that y is the result of the operation xe mod n. No, not really. It means the remainders mod n of both sides are the same. There is no guarantee that y < n at all. I think the confusion comes from the fact that mod n is actually not "part of the right hand side". Instead it applies to the whole expression. If you wanted to be "verbose" it would actually be: y mod n == xe mod n So while we write mod n only on one "side", it's something that applies equally to both sides.
Cornell Computer Science
cs.cornell.edu › courses › cs2800 › 2016sp › lectures › lec12-modular.html
Modular arithmetic (CS 2800, Spring 2016)
Definition: given an integer m, two integers a and b are congruent modulo m if m|(a − b). We write a ≡ b (mod m). I will also sometimes say equivalent modulo m. Notation note: we are using that "mod" symbol in two different ways. The first was defined in a previous lecture: a mod b denotes ...
Top answer 1 of 13
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- I am a human being.
- You are a human being.
Therefore, I am you: right ? Well, as it turns out, the answer is no. It simply means that we belong to the same class. Likewise, 
, but 
.
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I prefer to use the congruence exclusively (I view the remainder operation as a source of headache even though I realize that it is a necessary evil in computing).
But I use both equal (
) and congruent to (
) signs together when processing a lengthy calculation in modular arithmetic. My freshman algebra students quickly catch on with my calculations like
$$
12^{3004}\equiv5^{3004}=5^{3000}\cdot 5^4=(5^6)^{500}\cdot 5^4\equiv 1^{500}\cdot5^4=25^2\equiv4^2=16\equiv2\pmod7.
$
=$ when there is an equality of integers between the steps, and 
when I mean a congruence. The power of laws obeyed by congruences is apparent. Checking/following the progress of the calculation is easier this way. Of course, using 
all the way is correct also. The 
is there as a reminder that in this step we do something that only results in a congruence.
As the students become acquainted with the language of residue class rings, I gradually stop making the distinction between 
and 
as well as, clarity of context permitting, the distinction between 
and 
.
Thinking about what that would look like when done by somebody who is only familiar with binary mod makes me shudder.
Encyclopedia of Mathematics
encyclopediaofmath.org › wiki › Congruence
Congruence - Encyclopedia of Mathematics
If the difference $ a-b $ is not divisible by $ m $, then $ a $ and $ b $ are said to be incongruent modulo $ m $, and in order to express the incongruency of $ a $ and $ b $, the symbol · $$ a \ \not\equiv \ b \ ( \mathop{\rm mod}\nolimits \ m) $$ is used. The congruence $ a \equiv b \ ( \mathop{\rm mod}\nolimits \ m) $ expresses that $ a $ and $ b $ have identical remainders when divided by $ m $.
Expii
expii.com › t › what-is-congruence-modulo-n-3377
What Is Congruence Modulo N? - Expii
Two numbers are said to be congruent modulo N if their difference is divisible by N. Each integer belongs to one of N congruence (or residue) classes modulo N.
Csusm
public.csusm.edu › aitken_html › m372 › modulus1.pdf pdf
CONGRUENCE AND MODULUS: PART 1
that b % m is “b modulo m”). We call % the modulo operation, but perhaps a better name · for it is residue operation since it produces the residue not the modulus. This operation is · not popular with number theorists who tend to prefer using congruences ≡(see below). But · the symbol % is ...
Mathematics LibreTexts
math.libretexts.org › campus bookshelves › mount royal university › higher arithmetic › 3: modular arithmetic
3.1: Modulo Operation - Mathematics LibreTexts
November 22, 2024 - \(a\) is congruent to \(b\) modulo \(m\) denoted as \( a \equiv b (mod \, n) \), if \(a\) and \(b\) have the remainder when they are divided by \(n\), for \(a, b \in \mathbb{Z}\).
ScienceDirect
sciencedirect.com › topics › mathematics › congruence-modulo
Congruence Modulo - an overview | ScienceDirect Topics
Congruence of elements is defined here exactly as it is for rational integers, or, more generally, for elements of any ring (see the Supplement, Section 4.1): α ≡ β (mod γ) means that α – β is divisible by γ. If γ = pn — ε, where ε is a unit, then any congruence modulo γ is ...
K. Conrad
kconrad.math.uconn.edu › blurbs › ugradnumthy › modarith.pdf pdf
MODULAR ARITHMETIC KEITH CONRAD 1. Introduction
Example 2.5. Taking m = 2, every integer is congruent modulo 2 to exactly one of 0 and 1. Saying n ≡0 mod 2 means n = 2k for some integer k, so n is even, and saying n ≡1 mod 2 · means n = 2k + 1 for some integer k, so n is odd. We have a ≡b mod 2 precisely when a · and b have the same parity: both are even or both are odd. Example 2.6. Every integer is congruent mod 4 to exactly one of 0, 1, 2, or 3. Congruence
Physics Read
physicsread.com › home › how do you write congruence modulo(mod n) in latex?
How do you write congruence modulo(mod n) in LaTeX?
August 16, 2025 - Congruence modulo syntax will consist of two individual commands, \equiv and \mod commands.
Quora
quora.com › Geometry-uses-the-symbol-to-represent-congruent-Modulo-Arithmetic-uses-the-symbol-to-represent-equivalent-Do-the-symbols-mean-the-same-thing-Are-they-interchangeable
Geometry uses the symbol ≅ to represent 'congruent'. Modulo Arithmetic uses the symbol ≡ to represent 'equivalent'. Do the symbols mean the same thing? Are they interchangeable? - Quora
Answer (1 of 3): I’m confused by the American watered down use of the word congruent. It seems to mean equal. My understanding of congruent refers to shapes, particularly triangles, which are equal in all respects where one would fit exactly over the other.
Stack Overflow
stackoverflow.com › questions › 78535327 › modulo-arithmetic-modular-congruence
cryptography - (Modulo Arithmetic) Modular congruence - Stack Overflow
Formally, "calculating time" is described by the theory of congruences. We say that two integers are congruent modulo m if a ≡ b mod m.
UBC Math
personal.math.ubc.ca › ~PLP › book › sec-congruence.html
5.3 Congruence modulo \(n\)
Thus being congruent modulo 2 implies that they have the same parity. ... Now assume that \(a,b\) have the same parity. Then either they are both even or they are both odd. When \(a,b\) are both even, we can write \(a=2k, b=2\ell\) and so \(a-b = 2(k-\ell)\text{.}\) ... In both cases the difference \(a-b\) is divisible by 2 and so \(a \equiv b \mod 2\) as required. ... Perhaps the main reason that congruence modulo \(m\) is so important is that congruence interacts very nicely with basic arithmetic operations.
NYU Math
math.nyu.edu › ~hausner › congruence.pdf pdf
3 Congruence
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