UBC Math
personal.math.ubc.ca › ~PLP › book › sec-congruence.html
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 ...
Khan Academy
khanacademy.org › computing › computer-science › cryptography › modarithmetic › a › congruence-modulo
Congruence modulo (article) | Cryptography
We cannot provide a description for this page right now
Videos
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
5 days ago - After a wait of exactly 12 hours, the hour hand will be right where it started, so 12 acts as 0; one writes 12 ≡ 0 (mod 12). Given an integer m ≥ 1, called a modulus, two integers a and b are said to be congruent modulo m, if their difference a − b is an integer multiple of m; that is, ...
Vaia
vaia.com › all textbooks › math › discrete mathematics with applications › chapter 10 › problem 3
As in Example 10.1.2, the congruence modulo 2 relation - Math
We need to prove that for any even integer n, \(n E 0\). Let n be an even integer. By definition, we can write \(n = 2k\) for some integer \(k\). Now, according to the congruence modulo 2 relation, \(n E 0 \Leftrightarrow n-0\) is even. So, for n to be even, we need to prove that \(n-0\) is even.
American Institute of Mathematics
aimath.org › news › congruentnumbers › modulo.html
Congruence basics
We say integers a and b are "congruent modulo n" if their difference is a multiple of n. For example, 17 and 5 are congruent modulo 3 because 17 - 5 = 12 = 4⋅3, and 184 and 51 are congruent modulo 19 since 184 - 51 = 133 = 7⋅19. We often write this as 17 ≡ 5 mod 3 or 184 ≡ 51 mod 19.
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 ...
Whitman College
whitman.edu › mathematics › higher_math_online › section03.01.html
3.1 Congruence
Part 6: By hypothesis $n|a-b$ and $n|c-d$, so we have $n|(a-b)-(c-d)$. Rearranging the terms, this means $n|(a-c)-(b-d)$, so $a-c\equiv b-d$. Part 8: This follows from part 7, but it is easy to prove it directly: since $a\equiv b$, $n|a-b$. Therefore, $$ n|(a-b)(a^{j-1}+a^{j-2}b+…+ab^{j-2}+b^{j-1})=a^j-b^j, $$ so $a^j\equiv b^j$. Be sure you notice how often we have used lemma 3.1.2.$\qed$ Parts 5–8 can be summarized by saying that in any expression involving $+,-,\cdot$ and positive integer exponents (that is, any "polynomial''), if individual terms are replaced by other terms that are congruent to them modulo $n$, the resulting expression is congruent to the original.
Okstate
math.okstate.edu › people › binegar › 3613 › 3613-l11.pdf pdf
LECTURE 11 Congruence and Congruence Classes
it is clear that a ≡r (mod n). Thus, any integer a is congruent modulo n to some r ∈{0, 1, 2, . . . , n−1}. ... Definition 11.9. The set of all congruence classes modulo n is denoted Zn (which is read “Z mod n”).
UCI Mathematics
math.uci.edu › ~ndonalds › math180a › 3congruence.pdf pdf
3 Congruences and Congruence Equations
It appears that we were able to divide our original congruence by 6, but at the cost of dividing the · modulus by 2: it just so happens that 2 = gcd(6, 8)...
Carnegie Mellon University
math.cmu.edu › ~jmackey › summer › congruences.pdf pdf
21-128 Congruences Definitions of congruence
All these nice properties of congruence means that we can rearrange congruences just like we · rearrange equations provided all we do is add, subtract and multiply. ... So we can’t do division in modular arithmetic. But we almost can, at least, when a number is · relatively prime to the modulus. The feature of division that makes it useful in solving equations · is cancellation: if 2x = 4 then x = 2.
Ericrowland
ericrowland.github.io › investigations › modulararithmetic.html
Modular Arithmetic An Introduction
Here, the "≡" symbol is not equality but congruence, and the "mod 2" just signifies that our modulus is 2. The above statement is read "Zero plus zero is congruent to zero, modulo two." The statement "the sum of an even number and an odd number is odd" is represented by
Reddit
reddit.com › r/explainlikeimfive › eli5: how does a congruence modulo work?
r/explainlikeimfive on Reddit: ELI5: How does a congruence modulo work?
July 20, 2020 -
I think I understand the basics of the modulo function. Assuming modulo 3 when counting up you'd go 0 > 1 > 2 > 0 ...
But then I see this equation:
a ≡ b (mod n)
And have trouble understanding it. I'm pretty sure it means that a and b have a congruent relationship when a modulo function of n is applied to them. But I'm not sure if this is correct, and what a congruent relationship is.
Thanks!
Top answer 1 of 3
1
I'm pretty sure it means that a and b have a congruent relationship when a modulo function of n is applied to them Yes, that's correct. All that means is that if you take a and b and remove as many factors of n as possible from each one, you get the same thing. For example, 1 ≡ 3 (mod 2) and 9 ≡ 5 (mod 4).
2 of 3
1
So, a ≡ b (mod n) simply means that (a-b) is divisible by n. This is equivalent to saying that if you apply the modulo n function to both a and b then you get the same result. You can easily check and see that this relation has three properties: If a ≡ b (mod n) then b ≡ a (mod n), because b-a is also divisible by n. This is called symmetry. a ≡ a (mod n) for every a, because a-a is always divisible by n. This is called reflexivity. If a ≡ b (mod n) and b ≡ c (mod n) then a ≡ c (mod n), because c-a = (c-b)-(a-b) which is divisible by n. This is called transitivity. A relation that has these three properties is called a equivalence relation. Why? Because it means that if two elements are in relation to each other, then they are "equivalent" in regards to that relation. Another cool thing is that when you an equivalence relation, you can group the elements into equivalence classes - divide all the elements into sets so that two elements are in the same set if and only if they are equivalent. Now, the modulo relation has another cool property - it preserves the algebraic operations. If a ≡ b (mod n) and c ≡ d (mod n) then a+c ≡ b+d (mod n), and the same goes for multiplication. In this case, the relation is called a congruence relation. This means that if you want to calculate (a+b) (mod n) then you can calculate a (mod n) + b (mod n) and apply (mod n) to that, and you'll get the same result.
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.
Top answer 1 of 2
1
That depends on your definition of the remainder, which in turn depends on a definition of 'integer division'.
It's quite easy for positive numbers: the result of division is the largest integer not exceeding the exact result. For example 5/8 = 0. Then the remainder is 5–8*(8/5) = 5–8*0 = 5.
For negative numbers, however, a problem appears with a meaning of 'the largest'. One can assume it is the value largest with respect to its absolute value, i.e. the result is rounded towards zero (some programming languages work this way); then the integer division (–5)/8 results in –0=0, and the remainder is –5.
Or one can take literally the largest value, in which case (–5)/8=–1 and then the remainder is 3.
2 of 2
0
A quick answer is that when we work with modulo n and you are using the following definition:
two numbers, namely a and b, are congruent modulo n <=> a%n = b%n
We have to consider the same criteria in order for them to be equal, and that is, to consider a remainder of the same sign as the divisor.
The long answer involves some group theory in there. It is not easy to sumarize in a few words, but can be simply explained, using the example you have provided along the way. First, we will consider the group of remainders modulo, that is, a set of posible positive remainders when a integer is divided by n. Through the perspective of the group, -3 and 5 are the same element, because -3 + 8 = 5.
P.D.: I advise you from reading that book if such definition was given, such vague definitions are misleading and not rigorous in mathematics. By the way, if you want a good book about number theory I recommend: H. Rosen, Kenneth, Elementary Number Theory, Fifth Edition, Pearson ISBN-0-321-26314-6
K. Conrad
kconrad.math.uconn.edu › blurbs › ugradnumthy › modarithshort.pdf pdf
MODULAR ARITHMETIC (SHORT VERSION) 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 ...
University of Southampton
southampton.ac.uk › ~wright › 1001 › congruences.html
5.1 Congruences | MATH1001 Introduction to Number Theory
If we rearrange the digits then the two numbers have the same digit sum and therefore are congruent mod \(9\). Now if we examine the table of powers of \(2\) we see that \(2^0\equiv 1\), \(2^1\equiv 2\), \(2^2\equiv 4\), \(2^3\equiv 8\), \(2^4\equiv 7\), \(2^5\equiv 5\), \(2^6\equiv 1\) and that this pattern then cycles with period 6. So if two powers of \(2\) are congruent mod \(9\) then they are of the form \(2^n\) and \(2^{n+6k}\) for some \(k\). It follows that one of them is at least \(64\) times bigger than the other so they cannot have the same number of digits. A useful alternative definition of congruence \(\text{mod }(n)\) Lemma 5.7 ·
Mathematics LibreTexts
math.libretexts.org › bookshelves › combinatorics and discrete mathematics › elementary number theory (raji) › 3: congruences
3.1: Introduction to Congruences - Mathematics LibreTexts
July 7, 2021 - We say that \(a\) is congruent ... 5 (mod \ 7)\). Similarly \(2k+1 \equiv 1 (mod\ 2)\) which means every odd number is congruent to 1 modulo 2....
K. Conrad
kconrad.math.uconn.edu › math3240s20 › handouts › 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 ...
Omni Calculator
omnicalculator.com › math › modulo
Modulo Calculator
May 8, 2025 - Usually, when we use the word modulo, ... 3 equals 2 — so it's simply finding the remainder. In a strict definition, the modulo means: With respect to the specified modulus. ... A is the same as B modulo C, except for differences accounted for or explained by C. Which is the definition we wrote about in the congruence modulo ...
Ucalgary
cdm.ucalgary.ca › article › view › 77692
Congruence properties modulo powers of 2 for partition pairs into distinct parts | Contributions to Discrete Mathematics
October 28, 2025 - A result derived by Ray and Barman reveals that almost all values of $Q_2(n)$ are also divisible by $2^m$ with any fixed positive integer $m$. Quite recently, the author derived several internal congruences and congruences modulo powers of $2$ satisfied by $Q(n)$. In this paper, we prove some internal congruences and congruences modulo powers of 2 for $Q_2(n)$. Moreover, we prove an infinite family of congruence relations modulo $4$ and dozens of congruence relations modulo powers of $2$ enjoyed by $Q_2(n)$. Finally, we pose two conjectures on congruence properties modulo powers of $2$ for $Q_2(n)$.