### 数学代写|数论作业代写number theory代考|Math453

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|Euclid’s Algorithm

While finding the ged of two integers (not both 0 ), we can of course list all the common divisors and pick the greatest one amongst those. However, if $a$ and $b$ are very large integers, the process is very much time consuming. However, there is a far more efficient way of obtaining the gcd. That is known as the Euclid’s algorithm. This method essentially follows from the division algorithm for integers.
To prove the Euclidean algorithm, the following lemma will be helpful.
Lemma 2.4.1. If $a=q b+r$ then the $\operatorname{gcd}(a, b)=\operatorname{gcd}(b, r)$.
Proof. Let $d=\operatorname{gcd}(a, b)$ and $d_1=\operatorname{gcd}(b, r)$. Then, $d|a, d| b$ implies $d \mid(a-q b)$ 1.e, $d \mid r$. Thus $d$ is a common divisor of $b$ and $r$, hence $d \mid d_1$. Similarly, $d_1\left|b, d_1\right| r$ implies $d_1 \mid(b q+r)$ 1.e., $d_1$ divides both $a$ and $b$. Then, $d_1 \mid d$. Thus, $d=d_1$, as both $d$ and $d_1$ are positive by our definition of gcd.

Theorem 2.4.5. Euclid’s Algorithm: Let $a$ and $b(a>b)$ be any two integers If $r_1$ is the remainder when $a$ is divided by $b, r_2$ is the remainder when $b$ is divided by $r_1, r_3$ is the remainder when $r_1$ is divided by $r_2$ and so on. Thus $r_{n+1}=0$, then the last non zero remainder $r_n$ is the $\operatorname{gcd}(a, b)$.

Proof. Euclid’s algorithm is an efficient way of computing the ged of two integers by repeated application of the above lemma. At each step the size of the integers concerned gets reduced. Suppose we want to find the gcd of two integers $a$ and $b$, neither of them being 0 . As $\operatorname{gcd}(a, b)=\operatorname{gcd}(a,-b)=\operatorname{gcd}(-a, b)=\operatorname{gcd}(-a,-b)$, we may assume $a>b>0$. By performing division algorithm repeatedly, we obtain Theorem 2.4.5. Euclid’s Algorithm: Let $a$ and $b(a>b)$ be any two integers If $r_1$ is the remainder when $a$ is divided by $b, r_2$ is the remainder when $b$ is divided by $r_1, r_3$ is the remainder when $r_1$ is divided by $r_2$ and so on. Thus $r_{n+1}=0$, then the last non zero remainder $r_n$ is the $\operatorname{gcd}(a, b)$.

## 数学代写|数论作业代写number theory代考|Least Common Multiple

There is a concept parallel to that of the greatest common divisor of two integers, known as their least common multiple. Prime factorizations can also be used to find the smallest integer that is a multiple of two positive integers(treated in later chapters). The problem of finding this integer arises when fractions are added.

Definition 2.5.1. The least common multiple of two positive integers a and $b$ is the smallest positive integer that is divisible by a and $b$, denoted by lcm $(a, b)$ or $[a, b]$.
The above definition can also be formulated as follows:
Definition 2.5.2. The least common multiple of two nonzero integers a and $b$ is the positive integer $l$ satisfying the following:

1. $a \mid l$ and $b \mid l$.
2. If $a \mid c$ and $b \mid c$, with $c>0$, then $l \leq c$.
Example 2.5.1. We have the following least common multiple: $\operatorname{lcm}(16,20)=$ $80, \operatorname{lcm}(24,36)=72, \operatorname{lcm}(4,20)=20$, and $\operatorname{lcm}(5,13)=65$.

Remark 2.5.1. Given nonzero integers a and $b, \operatorname{lcm}(a, b)$ always exists and $\operatorname{lcm}(a, b)<|a b|($ Verify!).

Proposition 2.5.1. For nonzero integers $a$ and $b$, the following statements are equivalent(TFAE):

1. $\operatorname{gcd}(a, b)=|a|$.
2. $a \mid b$.
3. $\operatorname{lcm}(a, b)=|b|$.
Proof. (1) $\Rightarrow(2)$ : Let (1) holds. Then $\exists n \in \mathbb{Z}$ such that $b=|a| n$. Now $a>0 \Rightarrow$ $b=a n \Rightarrow a \mid b$. Again, $a<0 \Rightarrow|a|=-1 \Rightarrow b=(-a) n \Rightarrow b=a(-n) \Rightarrow a|b . a| b$. Hence (2) holds.

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Least Common Multiple

1. $a \mid l$ 和 $b \mid l$.
2. 如果 $a \mid c$ 和 $b \mid c$ ，和 $c>0$ ，然后 $l \leq c$. 示例 2.5.1。我们有以下最小公倍数： $\operatorname{lcm}(16,20)=$ $80, \operatorname{lcm}(24,36)=72, \operatorname{lcm}(4,20)=20$ ， 和lcm $(5,13)=65$.
备注 2.5.1。给定非零整数 $a$ 和 $b, \operatorname{lcm}(a, b)$ 总是存在并且 $\operatorname{cm}(a, b)<|a b|$ (核实! )。
提案 2.5.1。对于非零整数 $a$ 和 $b$ ，以下语句是等价的 (TFAE)：
3. $\operatorname{gcd}(a, b)=|a|$.
4. $a \mid b$.
5. $\operatorname{lcm}(a, b)=|b|$.
证明。(1) $\Rightarrow(2)$ : 让 (1) 成立。然后 $\exists n \in \mathbb{Z}$ 这样 $b=|a| n$. 现在 $a>0 \Rightarrow$ $b=a n \Rightarrow a \mid b$. 再次，
$a<0 \Rightarrow|a|=-1 \Rightarrow b=(-a) n \Rightarrow b=a(-n) \Rightarrow a|b . a| b$. 因此 (2) 成立。

## 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。