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## 数学代写|现代代数代写Modern Algebra代考|Modular Addition, Subtraction, and Multiplication

One of the steps in Gauss’s proof of the ruler-and-compass constructibility of the regular I7-sided polygon called for the verification of the identity
$$\left(\zeta+\zeta^{16}\right)\left(\zeta^{13}+\zeta^{4}\right)=\zeta^{14}+\zeta^{5}+\zeta^{29}+\zeta^{20}=\zeta^{14}+\zeta^{5}+\zeta^{12}+\zeta^{3}$$
In his writings Gauss used an abbreviation that replaced each $\zeta^{k}$ with the symbol $[k]$. Since $\zeta^{k+17}=\zeta^{k}$, it follows that, in Gauss’s notation, $[k+17]=[k]$ for each integer $k$. This is an example of modular arithmetic. For any positive integer $n$, the two integers $a$ and $b$ are said to be congruent modulo $n$, and we write
$$a=b \quad(\bmod n)$$
whenever $n$ is a divisor of $a-b$. This, by Corollary 2.17, is tantamount to saying that $\zeta^{a}=\zeta^{b}$ where $\zeta$ is any primitive $n$-th root of 1 . Thus, $7 \equiv 3(\bmod 4), 2 \equiv 14(\bmod 6)$, and $-3 \equiv 35(\bmod 19)$. Note that if $a \equiv a^{\prime}(\bmod n)$ and $b \equiv b^{\prime}(\bmod n)$, and if $\zeta$ is as above, then
$$\zeta^{a+b}=\zeta^{a} \zeta^{b}=\zeta^{a^{\prime}} \zeta^{b^{\prime}}=\zeta^{a^{\prime}+b^{\prime}}$$
and
$$\zeta^{a b}=\left(\zeta^{a}\right)^{b}=\left(\zeta^{a^{\prime}}\right)^{b}=\left(\zeta^{b}\right)^{a^{\prime}}=\left(\zeta^{b^{\prime}}\right)^{a^{\prime}}=\zeta^{a^{\prime} b^{\prime}}$$ and hence, $a+b \equiv a^{\prime}+b^{\prime}(\bmod n)$ and $a b \equiv a^{\prime} b^{\prime}(\bmod n)$.

## 数学代写|现代代数代写Modern Algebra代考|The Euclidean Algorithm and Modular Inverses

It turns out that the best way to deal with the question of invertible elements in $\mathbb{Z}_{n}$ involves the Euclidean algorithm for finding the grearest common divisor of two integers. This is a problem that was considered by many of the earliest mathematicians, including Euclid. An integer that is a divisor of both the integers $m$ and $n$ is said to be a common divisor. If such a common divisor of $m$ and $n$ has the additional property that it is divisible by all the common divisors of $m$ and $n$, then it is the greatest common divisor (GCD) or highest common factor (HCF) of $m$ and $n$ and it is denoted by $(m, n)$. Thus, $\pm 1$, $\pm 2, \pm 3, \pm 4, \pm 6$, and $\pm 12$ are all the common factors of 24 and 36 , but their $\mathrm{HCF}$ is 12 . Thus,
$$12=(24,36)=(-24,36)=(-24,-36) .$$
Note that since every integer divides 0 , it follows that $(0,0)$ does not exist. In Propositions I and 2 of Book vir of The Elements Euclid suggests the following method for finding the greatest common divisor of the two positive integers $m \geq n$. Suppose first that $n$ is a divisor of $m$. Then it is clear that $(m, n)=n$. If $n$ is not a divisor of $m$, then $n$ and $m-n$ are positive integers such that $(m-n, n)=(m, n)$. The reason for this is that every common divisor of $m$ and $n$ is clearly also a common divisor of $m-n$ and $n$, and, vice versa, every common divisor of $m-n$ and $n$ is also a divisor of $m=(m-n)+n$ and $n$. In other words, the set of common divisors of the pair ${m-n, n}$ is identical with the set of common divisors of the pair ${m, n}$. Consequently, the two pairs also have the same greatest common divisor.

# 现代代数代考

## 数学代写|现代代数代写Modern Algebra代考|Modular Addition, Subtraction, and Multiplication

$$\left(\zeta+\zeta^{16}\right)\left(\zeta^{13}+\zeta^{4}\right)=\zeta^{14}+\zeta^{5}+\zeta^{29}+\zeta^{20}=\zeta^{14}+\zeta^{5}+\zeta^{12}+\zeta^{3}$$

$$a=b \quad(\bmod n)$$

$$\zeta^{a+b}=\zeta^{a} \zeta^{b}=\zeta^{a^{\prime}} \zeta^{b^{\prime}}=\zeta^{a^{\prime}+b^{\prime}}$$

$$\zeta^{a b}=\left(\zeta^{a}\right)^{b}=\left(\zeta^{a^{\prime}}\right)^{b}=\left(\zeta^{b}\right)^{a^{\prime}}=\left(\zeta^{b^{\prime}}\right)^{a^{\prime}}=\zeta^{a^{\prime} b^{\prime}}$$

## 数学代写|现代代数代写Modern Algebra代考|The Euclidean Algorithm and Modular Inverses

$$12=(24,36)=(-24,36)=(-24,-36) .$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
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