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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Modular arithmetic, specifically n! mod m

Is there a theorem that makes solving $$ n! \equiv x \mod m $$ knowing that both $n$ and $m$ are prime? And if not, what would be the best way to go about finding $x$? cheers
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4answers
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Prove that $a$ and $a^{-1}$ inverse have the same order in $Z_n$

So there is a question in my lecture notes that I'm not too sure how to approach. It reads as follows: Suppose $a$ is invertible modulo $n$. Prove that $a$ and $a^{-1}$ have the same order in ...
102
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A variation of Fermat's little theorem

Fermat's little theorem states that for $n$ prime, $$ a^n \equiv a \pmod{n}. $$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
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1answer
24 views

Distributive modulo?

I would like to know if the modulo operation has distributivity like this: $$A+B+C \pmod{M} = (A+B)\pmod{M} +C \pmod{M}$$? Does the equality hold true?
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2answers
226 views

How can I solve $4x + 51y = 9$ using congruences?

I'm given: $4x+51y=9$. I am given a hint that when we use $4x=9 \pmod{51}$ we get $x = 15 + 15t$, and also if we use the congruence $51y=9 \pmod 4$ we get $y=3+4s$. They say it's handy to then find ...
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Find the lowest degree of the polynom $P$?

I have to determine the lowest degree of $P$ given by the following system : $\left\{ \begin{array}{l} P \equiv 2X \ \mod[X^2 -2X +1] \\ P \equiv 3X \ \mod[X^2 -4X+4] \end{array} \right.$ First, ...
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2answers
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Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$

Question: Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$ First note that when $p$ is prime and $1\leq r\leq p-1$ the binomial coefficient $$\binom{p}{r}=\frac{p!}{r!(p-r)!}$$ is ...
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1answer
445 views

Solving a non-linear congruence

How can we solve for $x$, knowing the integer $n$ and the real numbers $a$ and $b$, the following non-linear congruence: $(x+a)^2=-b\pmod{n}$ Specifically in this example: ...
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1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
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2answers
37 views

Given that $1 = 27 \times 11 - 74 \times 4$, solve the following equations in modulo $74$: $ 3x - y = 1$; $2x + 3y = 0$

Given that $1 = 27 \times 11 - 74 \times 4$, solve the following equations in modulo $74$: $3x - y = 1$; $2x + 3y = 0$. Thank you.
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If the euclidean algorithm is used to solve an equation ( i.e., $ax = b \mod(z)$) is the solution unique?

I have solved such an equation using the euclidean algorithm. However, unlike other methods, this gives one solution. Is this just one solution or the only solution. Help is much appreciated. Thank ...
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1answer
29 views

Modular arithmetic equivalent [on hold]

Let $A, B, C$ are integers from $1$ to $9$. In order to get an integer from this operation, A*B/C, I've devised two formulas. $(A*B) \pmod C$. $(A \pmod C)$ or $(B \pmod C)$. These two are not ...
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3answers
30 views

Simple mod 7 problem

I need to Show that $7x^3 + 2 = y^3$ has no solutions in integers x and y. The solution I am given is: Suppose there are solutions to this equation. Then mod 7 we have $2 ≡ y^3$ (mod $7$) and hence ...
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2answers
24 views

Sequence of perfect squares

Let $a,b\in \mathbb{N}$. Prove that, if $a$ is quadratic residue modulo $b$, then sequence $(a+kb)$, $k\in \mathbb{N}$, has infinite amount of perfect squares. How should I approach this ...
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109 views

Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$

Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$ Let $p$ be the least prime number such that $p\mid n$. And I want to show that $p=13$ Let $d$ be the least number such that: $14^d\equiv 0 ...
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3answers
46 views

Congruence $320 \equiv 1 (\text{mod }x)$ [on hold]

I have the following congruence $320 \equiv 1 (\text{mod }x)$ And the question is : find all the modulos $x$ that make this congruence true.
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1answer
27 views

Standard definition for $a$ being congurent to $b$ mod $n$

My text puts the definition for $$a\equiv b \bmod n$$ as $$n\mid(a-b).$$ On the other hand, certain sources puts the definition as $$n\mid(b-a).$$ Which exactly is the standard notation or is there a ...
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2answers
192 views

How to calculate a Modulo?

I really can't get my head around this "modulo" thing. Can someone show me a general step-by-step procedure on how I would be able to find out the 5 modulo 10, or 10 modulo 5. Also, what does this ...
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1answer
320 views

Modulus Cancellation Law

I'm trying to understand the proof for cancellation law in modulus which states that: ...
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5answers
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Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
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1answer
39 views

how to solve system of quadratic equations (mod N)

Given a two equations: $${(ax_1 + b)}^2 = c_1 \pmod N$$ $${(ax_2 + b)}^2 = c_2 \pmod N$$ $N=p.q$ $p$ and $q$ are large primes $x_1, x_2$ and $c_1, c_2$ are known Is it computationally feasible to ...
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Showing irreducibility of polynomials of degree 3 over the rationals

Let $\ g = X^3\ -9X + 16 $. Prove that $g$ is irreducible over the rational numbers. So far I have used reduction modulo $5$ and this gives $g_5 = X^3 +X + 1$. Then I get $$ g_5(0) \equiv 1 \pmod5,\\ ...
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$a^{13} \equiv a \bmod N$ - proof of maximum $N$

From Fermat's Little Theorem, we know that $a^{13} \equiv a \bmod 13$. Of course $a^{13} \equiv a \bmod p$ is also true for prime $p$ whenever $\phi(p) \mid 12$ - for example, $a^{13} = a^7\cdot a^6 ...
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System of linear congruence, not relatively prime

Consider we have the following set of congruences $$x\equiv b_i \pmod {m_i}$$ for all $1\leq i\leq d$. $m_i$'s doesn't have to be relatively prime, so the Chinese remainder theorem doesn't work here. ...
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2answers
30 views

Equality symbols in modular arithmetic

E.g., can I write $(a^{p})^{2p} \equiv a^{2p}=a^pa^p\equiv aa\equiv a^2\pmod{\! p}$? I often see equality symbols inbetween mod equivalences. The equality signs point out the equality is not ...
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1answer
398 views

What will be the multiplicative inverse of square root of 5 with respect to a natural number $M$?

Can such a number $N$ be found such that $\sqrt{5}N \equiv 1 \mod M$? If no,what can be the best approximation for $N$?
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1answer
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Examples of methods for solving modular equations

Simple mod questions. Can you show example to do such things? $x+40 \equiv 1 \pmod{88}$. $x \cdot 40 \equiv 1 \pmod{88}$. $5a+3b \equiv 1 \pmod{11}$ and $2a+b \equiv 7 \pmod{11}$. Thank you.
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Prove that $n^3=n \text{ mod }6$ for every integer $n$. [duplicate]

Prove that for every integer $n$ , $n^3=n \text{ mod }6$ I was having no clue how to do this, then I thought of case-by-case analysis and obviously it worked. The problem is that there were six case ...
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Proof of an congruence modulo n [duplicate]

I've the following theorem: For $n\in\Bbb Z$, prove that $n^3\equiv n \pmod{6}$ Please check whether I produced a good proof: 1) Let $k,n\in\Bbb Z$ s.t. $6=kn$ since $n^{3}$ is congruent to $n ...
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1answer
71 views

Solving to find the general equation with a “mod” equation

They probably aren't called "mod" equation but i couldn't think how else to word them, so I have this equation $8x + 10y ≡ 8 \pmod 7$ And have been tasked with finding the general solution, I know ...
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2answers
607 views

Prove that $x^3 \equiv x \bmod 6$ for all integers $x$

Prove that $x^3 \equiv x \bmod 6$ for all integers $x$ I think I got it, but is this proof correct? We can write any integer x in the form: $x = 6k, x = 6k + 1, x = 6k + 2, x = 6k + 3, x = 6k + ...
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Proof of a congruence relation

Let n∈N, and let a,b∈Z. Suppose that a≡b (mod n). Prove that n|a if and only if n|b. As can be proceed?
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Equivalence classes in $\mathbb{Z}_n$

I've the following exercise: Solve each of the following equations in the given set $\mathbb{Z}_n$: 1) $[5]+x=[1]$ in $\mathbb{Z}_9$ 2) $[2]\cdot x=[7]$ in $\mathbb{Z}_{11}$ For 1), is $x=5$ ...
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If $p$ is a prime number greater than $2$ and $k$ is a natural number so that $k<p$, how can I prove that?

If $p$ is a prime number greater than 2 and $k\in \mathbb{N}$ so that $k < p$, how can I prove that $p\choose k$ is congruent to $0 \bmod p$?
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Two real numbers which belong to distinct classes in the quotient group $\mathbb R/\mathbb Z$.

Let $x,y$ two real numbers. What does mean, in "pratical terms", that "$x,y$ belong to distinct classes in the quotient group $\mathbb R/\mathbb Z$"? Maybe that their difference $x-y$ isn't an integer ...
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3answers
54 views

On the distribution of multiples of 7 into intervals of length 11

Say we have two primes, say 7 and 11. We are to consider the positions of the multiples of 7 inside the (7 buckets of) multiples of $11$. So the buckets of 11 are: $[1,11],[12,22],\ldots ,[67,77]$, ...
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Sequence of real numbers which are distinct modulo 1.

Let $\{x_n\}_{n\in\mathbb N}$ a sequence of real numbers which are distinct modulo 1. Are the sequences $$\{\varepsilon x_n\}_{n\in\mathbb N} \ \ (\varepsilon\rightarrow 0), \ \ \ ...
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1answer
49 views

Show the congruence $x^{p-1}\equiv 1\pmod{p}$ has $p-1$ solutions

Question: Let $p$ be prime. show the congruence $x^{p-1}\equiv 1\pmod{p}$ has $p-1$ solutions Attempt: I know by Lagrange's theorem that this congruence will have at most $p-1$ solutions since $p-1$ ...
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3answers
116 views

Modular congruence, splitting a modulo

I can't find out, how to solve this. Will you give me some advice what to do in 4th step? Lot of thanks. This is my example: $7^{30}\equiv x\pmod{ 100}$ I want to compute it this way. These are my ...
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2answers
62 views

Number of fingers of a Martian

I have a question about what seems to be modular arithmetic, but I can't quite get the answer. The problem goes along the lines of: It is often said Earthlings use the decimal system because they ...
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1answer
61 views

What is $13^{498}$ (mod $997$)? [duplicate]

I have to determine $$13^{498} \pmod{997}$$ I know that it can only be $1$ or $-1$. But I don't quite know which. How can I decide?
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1answer
37 views

modulo RSA decrypt question

Given the following RSA generated public key: $P(3, 55)$. Which integer value should be chosen for $d$ to decrypt messages encrypted with $P$? Check your answer with $M = 8$ and $C = 17$. ...
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1answer
53 views

Modulus differentiation

For a Java project, I need to find a way to compute the derivate of a modulus function like $$f(x) = g(x) \pmod{h(x)}$$ for any value of $x$. I know that the modulus function is discontinuous. If ...
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Modular arithmetic of rotation a triange

A triangle is an n-gon with 3 rotational symmetries and 3 reflection symmetries. Each rotation is 120 degrees. Suppose the triangle begins with initial angle 240 degrees and rotates through 240 ...
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1answer
43 views

How to solve congruence modulo equations?

While studying Affine Cipher in cryptography it tells that we need to solve a system of modulo congruence equations. The equations are: $8\alpha+\beta\equiv 15 \pmod{26}$ $5\alpha+\beta\equiv 16 ...
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Show that if $a, b$ and $m$ are integers such that $m \geq 2$ and $a \equiv b \pmod{m}$, then $\gcd(a, m) = \gcd(b, m)$

Problem 1 (#3.5.32). Show that if $a, b$, and $m$ are integers such that $m \geq 2$ and $a \equiv b \pmod {m}$, then $\gcd(a, m) = gcd(b, m)$. Proof. Let $d = \gcd(a, m)$ Then $d \mid a$ and $d ...
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1answer
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Hensel's lemma modular arithmatic example problem

In an example for Hensel's Lemma we have met the criteria to use Hensel's lemma and have begun to apply it in a Hensel's iteration. We have $f(x)=x^2+1$ and our initial $x_0=2$ is a solution ...
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Prove that if $\{1^5,2^5,\ldots, (pq)^5\}$ is a complete residue system mod $pq$, then $\{1^5,2^5,\ldots,p^5\}$ is a complete residue system mod $p$.

Let $p,q\ge 2$ be coprime positive integers. Prove that if $\{1^5,2^5,\ldots, (pq)^5\}$ is a complete residue system mod $pq$, then $\{1^5,2^5,\ldots,p^5\}$ is a complete residue system mod $p$. ...
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Why isn't Euler's theorem working to find the smallest $k$ such that $10^k \equiv 1 \pmod {\!9}$?

$10^k \equiv 1 \pmod {\!9}$ According to Euler's theorem and the Carmichael function, smallest $k$ is $\phi(9) = 6$, but clearly the smallest $k$ is $k=1$. What am I doing wrong?
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6answers
41 views

Finding the inverse of a number under a certain modulus

How does one get the inverse of 7 modulo 11? I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure. Likewise, I have the same problem finding the inverse of 3 ...