Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $b-a$.

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Remainder when $11^{2402}$ is divided by $3000$? [closed]

What is the remainder when $11^{2402}$ is divided by $3000$? I just came across this question. I am a beginner in number theory. Your help would mean a lot.Thanks!!
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5answers
83 views

How I can find the result of $1761^3 \bmod 7$?

I would like to know how I can find the result of $1761^3 \bmod 7$. Is there any rule? Thanks so much for your help!
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2answers
119 views

Sum of two squares modulo p

I have heard somewhere that for all primes $p$, for all $k$, there exist $x, y$ s.t. $x^2 + y^2\equiv k \pmod{p}$? I recall that the proof is very elementary, but I can't remember such a proof. How ...
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71 views

Show that $a^2 \bmod b = (a \bmod b)^2 \bmod b$

Show that $$a^2 \mathrm{\ mod \ } b = (a \mathrm{\ mod \ } b)^2 \mathrm{\ mod\ } b$$ for $ a, b \in \mathbb{Z}^+ $. this was derived from an Informatics olympiad question.
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Help with Understanding Congruence Statement in Divisibility Proof

I am trying to understand the proof of a divisibility rule from this website. I've had very little exposure to modular arithmetic, so in order to attempt to understand the proof I spent the afternoon ...
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92 views

How to solve this modular equation? $x^{19} \equiv 36 \mod 97$.

How to solve the following? $x^{19} \equiv 36 \mod 97$. I am having trouble figuring this out. Which technique do I need to use? Chinese Remainder or Fermat's Little Theorem?
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4answers
47 views

how do you find the modular inverse

I need to find out the modular inverse of 5(mod 11), I know the answer is 9 and got the following so far and don't understand how to than get the answer. I know how to get the answer for a larger one ...
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25 views

In $Z[x]$, which pairs of $a,b$ commute as modulo operators, such that ($c$ mod $a$) mod $b =$ ($c$ mod $b$) mod $a$ for every $c$?

In $Z[x]$, which pairs of $a,b$ commute as modulo operators, such that ($c$ mod $a$) mod $b =$ ($c$ mod $b$) mod $a$ for every $c$? What I got so far is: Clearly the equasion holds for every pair ...
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0answers
77 views

Average and Standard Deviation for Angular Values

I'm trying to characterize an inexpensive accelerometer that provides (roll, pitch, yaw) values by generating basic running statistics (mean and standard deviation). But I'm unfamiliar with ...
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1answer
23 views

Negative number modular positive number ?

I want to understand how 1-%5 = 4 ? I already know that 1%5 = 1 and 2/5=2 and so on. but please explain this when is is negative as the previous example
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1answer
41 views

Digit wise modulo for calculating power function for very very large positive integers

I am writing a code to calculate $P^Q$ where $P$, $Q$ are positive integers which can have number of digits up to $100000$. I want the result as $r = P^Q \pmod{10^9+7}$, where $10^9+7$ is a prime ...
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54 views

Non cyclic subgroup of $\mathbb{Z}_5$

Let $H = \{0, 2, 3\} \subset \mathbb{Z}_5$. $H$ is a subgroup of $\mathbb{Z}_5$, since it is closed with respect to addition and with respect to inverses. Given that $\langle 2 \rangle = ...
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0answers
23 views

Testing for Linear Independence/rank mod m

I am working on cracking a hill cypher using modular linear algebra. Every example I have found online makes a big assumption that is not necessarily the case, and as I see it leaves a lot to be ...
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1answer
38 views

Proof for a relation

Define a relationship $R$ on $\mathbb{Z}$ by declaring that $xRy$ if and only if $x^2 \equiv y^2 (mod 4)$. Prove that $R$ is reflexive, symmetric and transitive. I'm unsure of where to start with ...
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163 views

Why didn't Fermat provide proofs of his theorems?

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, ...
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50 views

Modulo operation. How to prove it?

I noticed relation between modulo operation and number which is power of two Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to $ ( 3431242341 \mod 2^9 ) ...
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2answers
98 views

Binomial Congruence Modulo a Prime

Let $p$ be a prime and $a, b$ natural numbers such that $1 \leq b \leq a$. I am trying to prove that $$\binom{ap}{bp} \equiv \binom{a}{b} \pmod p.$$ Furthermore, I have been tasked with proving that a ...
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41 views

Given $m^2\bmod A$ and $m^2\bmod B$, how do you find $m^2 \bmod AB$?

Given $m^2\bmod A$ and $m^2\bmod B$, how do you find $m^2 \bmod AB$? I'm 99% sure this is an application of the Chinese remainder theorem, although my workings do not quite show how it can be ...
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20 views

Simplifying the expression involving mods

I'm trying to simplify the following expression (I hope to be able to write it in a nicer form) but I cannot: For $m \in \mathbb{N}$ and $n \in \mathbb{N}$, $l(m)$ is defined as \begin{equation*} l(m) ...
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5answers
90 views

finding mod of an expression with variables

I've been asked to show that: $n(n+1)(2n+1) \equiv 0 (\bmod 6)$ I found in a previous question that: $n(n+1)$ was divisible by $2$ and resulted in an even number e.g $n(n+1) \equiv 0 (\mod 2)$ so ...
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28 views

Converting infinite base-k expansions into base-j expansions

I understood the method of transforming a finite sized base-k numbers to another base (j) through the use of successive divisions For example $$12_{10} = 12/2 + 0*2^0 = 6/2 + 0*2^1 + 0*2^0 = 3/2 + ...
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49 views

Why 3 is a multiplicative inverse of 7 in modular arithmetic?

why is 3 a multiplicative inverse of 7 in modular arithmetic of 5 ? I'm not able to understand how this is true. PS: I know 3*7-1 % 5 = 0. I'm not able to make sense of inverses in modular ...
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96 views

“Remainder” operation in mod 2^32

I debated posting this here, in the cryptography SE, or the programming SE. Obviously, I chose here, but I'm not confident in my choice... I'm attempting to "undo" a function, but I've hit a slight ...
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18 views

Intersections of moduli

How do you find "intersection"s of moduli? For example what is the fastest way to find a number $n$ based on modular properties of $n$? Ex: Find is the first number $n$ such that: $n \equiv 2 \mod ...
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41 views

What is the remainder of this big number without doing major calculations?

I am solving a problem and came across a situation where to calculate remainder for big values with out doing major calculation. In my case I need to compute the expression: $$2^{n}-1+k ...
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1answer
31 views

Prove $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism.

I'm working on proving the following claim: "Let $r \in U(n)$ and $\forall s \in \mathbb{Z_n}$, $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism." ...
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What is remainder when $5^6 - 3^6$ is divided by $2^3$ (method)

I want to know the method through which I can determine the answers of questions like above mentioned one. PS : The numbers are just for example. There may be the same question for BIG numbers. ...
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1answer
43 views

Modulo Quadratic Polynomials

Can you, given a large number N, find a, b, c such that ax^2 + bx + c = 0 has at least N roots? All of this is in any mod you choose.
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74 views

Prove that $n^2 + 1$ is not a multiple of $6$ for any positive integer $n$

Prove that $n^2+1$ is not a multiple of $6$ for any positive integer $n$. I i think prime factorization would be a good way to go about this problem but I need some help.
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33 views

Prove if $n \equiv 2 \pmod 7$, then $7 \mid (n^2 + 10)$

Prove if $n \equiv 2 \pmod 7$, then $7 \mid (n^2 + 10)$. I tried saying since $n \equiv 2 \pmod 7$, then $7 \mid n - 2$. Thus $7 \mid -5( n - 2)$ or $7 \mid -5n + 10$ and $-5n \equiv 10 \pmod ...
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An analogue of Hensel's lifting for Fibonacci numbers

In this question Oleg567 conjectured the following interesting fact about Fibonacci numbers: $$ k\mid F_n\quad\Longrightarrow\quad k^d\mid F_{k^{d-1}n} $$ that can be regarder as an analogue of the ...
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Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
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131 views

Congruence modulo primes or in a polynomial ring over ${\rm GF}(2)$

Let $p, q$ be primes. Then the linear congruence $$ap \equiv r\pmod q$$ can be solved for $a\in\mathbb Z$ and will have a unique solution for each value of $r$ such that $0\leqslant a<q$. Am I ...
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45 views

Show that if $a^h ≡ 1\mod p$ then $ a^{ph} ≡ 1 \ \mod p^2$.

I don't know how to proceed. I know that regardless of what h is, it divides the order of a modulo $p$. I also know that the order of a divides $\phi(p) \ \text{mod} \ p$, where $\phi$ is Euler's ...
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x≡1 (mod 8) x≡9 (mod 12) has solution x = x_0. How many solutions mod 24 are there to the system of congruences?

Say, x≡1 (mod 8) x≡9 (mod 12) has solution x = x_0. How many solutions mod 24 are there to the system of congruences? I am worried this question is too easy to be true. That is why I am confused. ...
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4answers
53 views

Direct proof using modular arithmetic

Give a direct proof of $8\mid (3^n + 5^n)$ for all odd natural numbers. I know how to prove this by induction, I am not sure how to go about it using a direct proof. I would start by saying that ...
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1answer
46 views

Congruence and percentage

Suppose I have three statements of congruence: x = a mod n, y = b mod m, z = c mod p; Furthermore, x is a given percent of x + y + z, as is y and z. Does this uniquely determine x, y, z? Or does it ...
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37 views

Proof for a non-conditional statement

I'm having a bit of trouble doing this proof. If $a\in\mathbb{Z}$, then $a^3 \equiv a \pmod 3$. I know how to do proofs if there were conditional statements but not sure how to prove this with ...
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37 views

modulo group defined by an algebraic relation

I am asked if $\{n, n^{2}, n^{3}\}$ forms a group under multiplication modulo $m$ where $m = n + n^{2} + n^{3}.$ As an example we see that $\{2, 4, 8\}$ does form a group modulo $14,$ with identity ...
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3answers
69 views

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that ...
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61 views

Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
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24 views

Keep a Function Positive via Mod

I have a function $F$ and I want it to remain positive--i.e., $$- F=F,\quad F=F $$ Would sticking a $\mod 2$ in front of $F$ do this? That is, because $$-1\mod 2=1\mod 2=1 $$ Then let ...
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61 views

How to solve this $ (7/2)\bmod5$?

I know its answer is $3\cdot5$ but I want to ask that is the following true- $$(a/b)\bmod(p) = (a\bmod(p))\cdot((1/b)\bmod(p)))\bmod(p)$$ (where $a$ and $b$ are any integers and $p$ is a prime ...
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1answer
196 views

Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
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4answers
160 views

Is it possible to do modulo of a fraction

I am trying to figure out how to take the modulo of a fraction. For example: 1/2 mod 3. When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?
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53 views

A homework question, finding the maximum possible value of the sum of two remainders

If $a<b$ what is the maximum possible value of a mod b+ b mod a. I tried several times, the answer always came out to be 2a-2. But then it is not a choice. Am I right?
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1answer
60 views

PowerMod: Solving for the base

Given the problem $c^d \mod n = m$ and values for $d$, $n$, and $m$, how would one solve for $c$? A general solution or approach would be fine, as well as the values for my specific problem are as ...
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5answers
70 views

How many integers are there between 50 and 250 inclusive which are congruent to 1 mod 7?

Number of integers between $50$ and $250$ inclusive which are congurent to $1$ mod $7$. I understand that one could find the smallest and largest numbers in the interval $[50,250]$ that are congruent ...
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2answers
37 views

Tricks for Find Modular Inverses

I know that you can apply Euclid's Extended Algorithm, but I was wondering if there were "tricks" for guessing modular inverses. For example, if you have something like $ 13 \pmod{25}$ then you easily ...
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1answer
50 views

Solving Equations in $\mathbf{Z}/n\mathbf{Z}$ with Indices

Consider the equation $x^4 = 7,$ which we wish to solve in $\mathbf{Z}/29\mathbf{Z}.$ I was taught a technique for solving this problem, but I can't understand it. I'll try my best to describe it, ...