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Questions tagged [chinese-remainder-theorem]

For questions related to the Chinese Remainder Theorem and its applications.

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How many solutions does $x \equiv x^{-1} \pmod n$ have?

How many solutions does $x \equiv x^{-1} \pmod n$ have? $n$ is defined to be a positive integer, What I believe the solution will be is along the lines of 2 cases: When $n = 1$, the set of ...
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Some questions about $f: \Bbb Z_{44100} \rightarrow \Bbb Z_{150}\times\Bbb Z_{294}$

Given the function: $f: \Bbb Z_{44100} \rightarrow \Bbb Z_{150}\times\Bbb Z_{294}$ defined as follows $[x]_{44100} \rightarrow ([x]_{150},[x]_{294})$ Calculate $f(12345)$ - Answered A preimage of (...
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Composition of well defined maps is well defined?

If we have two mappings; $a \:mod \:NM \to a \:mod \: M $ and $a \: mod \: NM \to a \:mod \:N $ which are both well defined. Can we then conclude that the mapping $a \: mod \: NM \to (a ...
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Algorithm for finding the smallest integer that satisfies several modular congruence conditions?

my first question! I work a lot with numbers (finance) but very much an amateur mathematician - please be gentle. I have the following problem that has come from discussions about cryptography: Given ...
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System of (non-linear) congruence equations

I got a system of two congruence equations where one of them is non-linear. \begin{cases} 2*x^2 + 5 \equiv 4\ (\textrm{mod}\ 11) \\ x \equiv 3\ (\textrm{mod}\ 13) \end{cases} My idea was to rewrite ...
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exist $x$ such that $x^k \equiv m$ mod $(p_1\cdot p_2) \Leftrightarrow $ exists $x_1,x_2$ : $x_1^k\equiv m(p_1)$ and $x_2^k\equiv m(p_2)$

Let $p_1,p_2$ prime numbers, I wish to show that: exist $x$ such that $x^k \equiv m$ mod $(p_1\cdot p_2) \Leftrightarrow $ exists $x_1,x_2$ : $x_1^k\equiv m(p_1)$ and $x_2^k\equiv m(p_2)$ A first ...
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whats is the remainder when $38^{33^{41}}$ is divided by $11$?

What kind approach is to solve this kind of questions? I have already tried break down and squaring. But still stuck.
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4answers
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How to solve such a quadratic congruence equation?

I have the following equation: $y^2 \equiv r^2 \pmod n $ I know the values of y and n, I just need to find the values of r. Assuming that $y = 12654$ and $n = 79061$, my working is as follows: $ ...
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$x^2\equiv 5 \pmod{1331p^3}$

Let $p$ be given by $p=2^{89}-1$ and note that it is a Mersenne Prime. The problem is to find the number of incongruent solutions to $$ x^2\equiv 5 \pmod{1331p^3} $$ I began the problem by splitting ...
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Congruence involving CRT

I was working on a problem, I arrived at the point at which I have to find $17^{{{17}^{17}}^{17}} \pmod {25}$ My attempt: $$ 17^{{{17}^{17}}^{17}}\equiv 17^{{{{17}^{17}}^{17}} \pmod{\phi(25)}} \pmod {...
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Derived Chinese Remainder Theorem question

If we know an example of the CRT, e.g.: What $n$ is $3\pmod 7$ and $5\pmod{11}$? and we know the answer, in this case $n=38$, is it any easier to find the answer to a related CRT question, for ...
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The first digit and the last three digits of tower of exponents

How to find the first digit and the last three digits of ${{{{2}^{3}}^{4}}^{\cdots }}^{1000}$, where the expression contains all integer numbers (from $2$ to $1000$, in order)?
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Find the remainder of the division of polynomials

$x^{2007}$ divided by $x^2-x+1$. I consider to solve this problem, should I break the $x^{2007}$ to find the formula $x^2-x+1$?
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Is there a way to reverse the Chinese Remainder Theorem? What extra information do we need?

Dear math stackexchange community, Given a list of numbers < N, after generating the Chinese Remainder, is there a way to get back to the same list of numbers? Example: N = 100 List of numbers =...
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Chinese remainder theorem and Diophantine equation implementation

I needed an advise on implementing and solving one problem and the others like. I came across two sentences that I think will be helpful. Those are the Chinese remainder theorem and the Diofantic ...
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Modular incongruences and Chinese Remainder Theorem

Is there a way to adapt the Chinese remainder theorem to solve a series of modular incongruences, e.g. $n \not\equiv 0\ (\textrm{mod } 5)$ $n \not\equiv 0\ (\textrm{mod } 6)$ $n \not\equiv 0\ (\...
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Remainder for equation with multi-variable

I was trying to figure out the remiander when $(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$ is devided by $(a+b+c)^3-a^3-b^3-c^3$. I first tried to factor them but this guy don't factor in any nice way. I ...
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4answers
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Proving that $\mathbb{Z}/128\mathbb{Z}$ has exactly one maximal ideal?

I would like the prove that $\mathbb{Z}/128\mathbb{Z}$ has exactly one maximal ideal. I believe this has to do with the fact that $128 = 2^7$, but I'm a little lost on everything else here. I'm ...
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Chinese remainder theorem to find $1030^{989}\bmod\; 3003$?

so this is a slightly different take on a question I asked, but instead of the product of two numbers- this time it is a very large number raised to a very large power. I am meant to use the Chinese ...
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How many elements in $S$ satisfy $x^2=49 \pmod{5400}$?

Let $S=\{0,1,2,...,5399\}$. How many elements in $S$ satisfy $x^2=49 \pmod{5400}$? So I'm thinking about using Chinese remainder Theorem, but since $5400$ has many factors, wouldn't cracking it down ...
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Modular Arithmetic - Solve $a$ and $b$ given equation for $x$

This is an application to the Chinese Remainder Theorem. Given $x=63k+12\ \ \forall k\in\mathbb{Z}$, solve for $a$ and $b$ that satisfy the relation $$x\equiv a \mod7 \\ x\equiv b \mod 9$$ So far ...
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Find all solutions to the system $x \equiv 1 \pmod 4, x \equiv 0 \pmod 3$, and $x \equiv 5 \pmod 7$

Find all solutions to the congruences x $\equiv$ 1 (mod 4), x $\equiv$ 0 (mod 3), and x $\equiv$ 5 (mod 7). I got $M =m_1 * m_2 * m_3 = 4*3*7 = 84$ $M_1 = 21, M_2 = 28, M_3 = 12$ So I get $x = 21*u +...
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Use the Chinese Reminder Theorem to find; (1030 ∗ 989) mod 3003.

I have found a lot of help online for solving a system of congruences but I am not sure what to do with this? I am having a lot of trouble with the Chinese Remainder Theorem in general, but this one ...
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Modular solvability paired quadratic

We have $$B(x+y-L_1)y\equiv Mx\bmod A$$ $$A(x+y-L_2)x\equiv My\bmod B$$ where $A,B$ are known coprime in $[n,2n]$ and $M,L_1,L_2$ are known integers and we have $|x|,|y|<n$. Is there an effective ...
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Validation for a conjecture about Chinese Remainder Theorem for groups

I was wondering if the following statement is true: Let $G$ be a group with normal subgroups $H_1,H_2,...H_n$. Suppose $H_iH_j=G$ for all $i\neq j$. Then $G/H_1\cap H_2...\cap H_n\cong G/H_1 \times......
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33 views

Why converted values from Decimal to binary isn't the same? [closed]

the professor told us today about binary and decimal and how to convert them , and give us example of a decimal number (13) and we converted it to binary which is (1101) . Now when I'm trying to do ...
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Chinese Remainder Theorem on polynomials

Let $f(x) = x^5 + 3x^2 + 4$. Find all solution to the congruence. $f(x)\equiv 0 \pmod{12}$. I don't understand how to apply CRT on polynomials.
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How to tell if system of congruences where each base is a power of prime $p$ has a solution

$p$ is a prime number. How to tell if a system of congruences: \begin{align} x &\equiv a_1 \pmod{p^{i_1}} \\ x&\equiv a_2 \pmod{p^{i_2}} \\ &\dots\\ x &\equiv a_n \pmod{p^{i_n}} \...
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1answer
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Application of Euler's totient function to find last digits

Q:what are the last five digits of the number $2018^{2017^{.^{.^{.^{2^{1}}}}}}$. My Approach:I know how to find the last two digits of $N=2018^{2017^{k}} $ by$N=2018^{2017^{k}\pmod{\phi(25)}}\pmod {25}...
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What is the remainder of [closed]

$\dfrac{33^{100}}{50}$ I have done $\dfrac{2×{33^{100}}}{100}$ But it is still complex. How can we calculate the remainder of a number divided by 100?
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Simple number theory questions (CRT)

I have been watching the Chinese Remainder Theorem, and have some questions about things related to it. 1) I have some questions about the inverse in modular arithmetic. I think that it's easy to ...
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Problem regarding linear congruence

Which of the following statement is False ? 1) There exists a natural number which when divided by 3 leaves remainder 1 and which when divided by 4 leaves remainder 0 2) There exists a natural ...
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Is this a simple LCM or Ad hoc problem or modular arithmetic problem?

I've got a Kindergarten problem, which is to find smallest number which will be satisfied following condition: If we divide that number with $4, 6$ and $10$ then $2, 4$ and $8$ will be remainder ...
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Chinese remainder theorem in a Dedekind domain.

I want to prove the following : Let $a_1,...,a_n$ ideals and $x_1,...,x_n$ elements of a Dedekind domain $A$. Then the system of congruences $x \equiv x_i \pmod{a_i}$ for $i = 1,...,n$ has a solution $...
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138 views

Computer arithmetic on large integers, (Chinese remainder theorem)

When doing a problem on computer arithmetic with large integers, I reached a step when I needed to solve system of congruences. I came up with the following equations: $$x ≡ 65 \pmod{99})\\ x ≡ ...
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Find $x \in \mathbb{Q}(i)$ with $ |x - 1|_{2+i} < \frac{1}{\sqrt{5}} $, $|x+1|_{2-i} < \frac{1}{\sqrt{5}}$ and $|x|_{7} < \frac{1}{7} $

I wanted to try some examples with adeles and strong aproximation. Let $\mathfrak{p}_1 = 2+i$ and $\mathfrak{p}_2 = 2-i$ and $\mathfrak{p}_3 = 7$. Can we a single number $x \in \mathbb{Q}(i)$ that's ...
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Are we allowed to interchange product and inverse limits?

Currently, I am trying to show that the profinite completion $\hat{\mathbb{Z}}$ of $\mathbb{Z}$ is isomorphic to $\prod_p \mathbb{Z}_p$ (as topological groups) where $p$ runs through all prime numbers ...
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Number theory problem on Chinese remainder theorem

Let $T$ be the smallest positive integer which, when divided by $11,13,15$ leaves remainders in the sets $\{7,8,9\},\{1,2,3\},\{4,5,6\}$ respectively. What is the sum of the squares of the digits of $...
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Chinese remainder theorem - A, B and C think of a code

A, B and C are trying to think of a code. A remembers that after dividing by 13, the residue is 8. its double increased by 1234 after dividing by 17 leaves us residue of 7. C remembers that its triple ...
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Which of the following intervals contains integers satisfying the following three congruences:

Question: Which of the following intervals contains integers satisfying the following three congruences: $x\equiv 2\pmod 5, x\equiv 3\pmod 7$ and $x\equiv 4\pmod {11}$, (i) $[401,600]$, (ii) $[601,800]...
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Natural number that has a remainder of $1, 2, 3, 4$ respectively after dividing… [duplicate]

A number when divided by 2 leaves a remainder of 1. When it is divided by 3 leaves a remainder 2. When it is divided by 4 it leaves a remainder of 3. And when it is divided by 5 it leaves remainder of ...
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2answers
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Linear congruence equations

I'm trying to prove that the following system of congruence equations has a solution: $X \equiv 2 $ (mod $5^N$) $X \equiv 1 $ (mod $7^N$) $X \equiv 4 $ (mod $6^N-4$) being $N$ an integer number, $...
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Chinese Remainder Theorem: four square roots of 1 modulo N

Given an odd composite number $N$, where $N$ is not a prime power, I read the following in a Wikipedia article: As a consequence of the Chinese remainder theorem, the number $1$ has at least four ...
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Solving a system of congruent equations [closed]

1.$$10x\equiv 34 \pmod{63}$$ 2.$$11x\equiv 44 \pmod{64}$$ 3.$$12x\equiv 54 \pmod{65}$$ How am I supposed to solve it? I know that use of the Chinese remainder theorem is not allowed in this case ...
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Bezout identity on polynomial ring $\Bbb Q[x]$

Question: After proving that it exists, find $f(x)\in\Bbb Q[x]$ of degree at most 2 such that $$f(x)\equiv2 \text{ mod }x+1\ \text{and }\ f(x)\equiv x+1\text{ mod }x^2+1$$ My attempt: Since $(x+1)$...
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1answer
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Remainder of trinomial

What will be the remainder if I divide $(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$ by $(a+b+c)^3-a^3-b^3-c^3$. I have tried trinomial expention. But its still too big for long division. Is there a shorter ...
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3answers
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Consider this system of congruence equations.

\begin{cases} 4x \equiv 14 \pmod m \\ 3x \equiv 2 \pmod 5 \end{cases} I want to prove that for $m \in 4\mathbb{Z}$ there are no solutions(1). Moreover, I want to determine all m for which I have ...
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Modular arithmetic problem from Chinese remainder Theorem

Take $p,q$ to be coprimes with $p<q<2p$. If $w,x,y,z$ are unknowns such that $w,x,y,z<\min(p,q)$ and we know $(wy+xz)\bmod (p+q)$, $(wz+xy)\bmod (p+q)$ $wy\bmod q$ and $xz\bmod p$ then are $w,...
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1answer
112 views

Chinese Remainder Theorem, Miller-Rabin Primality test, and more…

Good day, I was going over the proof of the Miller-Rabin Primality Test and have a few questions regarding it. THE BOOK IS COMPLEXITY AND CRYPTOGRAPHY: AN INTRODUCTION Where $B_t = \{a\in\Bbb Z_n^...
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Direct sum isomorphism question

I found this problem and I'm not sure if I can apply the reminder theorem on it. Are $\mathbb{Z}_8\oplus\mathbb{Z}_{18}$ and $\mathbb{Z}_2\oplus\mathbb{Z}_{72}$ isomorphic? I tend to believe so, but I ...