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# 滑铁卢数学竞赛代考Waterloo Math Contest代考|Number theory

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## 滑铁卢数学竞赛代考Waterloo Math Contest代考|Number theory

By the end of this topic you should be able to:
(i) Use the tests of divisibility as a problem solving tool.
(ii) Use the congruence notation and do congruence arithmetic.
(iii) Use Fermat’s Little Theorem and Wilson’s theorem.
(iv) Use the Unique Factorization Theorem (also known as the Fundamental Theorem of Arithmetic).
(v) Use the Chinese Remainder Theorem.
As appetizers, here are two typical problems of the kind you should be able to solve when you have worked through Sections $1-4$, and might find even easier after Section 5 . You are invited to try them as soon as you wish. The solution of each is given at the end of Section 4, and a streamlined approach to Appetizer Problem 1 is given at the end of Section 5.

Appetizer Problem 1: Find the remainder when $2^{4901}$ is divided by 11.
(A) 1 (B) 3
(C) 6
(D) 10
(E) 2
Appetizer Problem 2: What are the last two digits in the number $11^{111}$ ?
(A) 01
(B) 11
(C) 21
(D) 31
(E) 41

## 滑铁卢数学竞赛代考Waterloo Math Contest代考|Divisibility, primes and factorization

We collect here some simple facts about divisibility that will be used frequently. In talking about divisibility of numbers, they are always understood to be integers. The idea of divisibility and the notation is introduced at the beginning of Toolchest 3 .

We say that $b$ divides $a$, and we write $b \mid a$, if there is an integer $c$ such that $a=b c$. Thus, $4 \mid 36$ because $36=4 \cdot 9$; and $(-3) \mid(18)$ because $18=$ $(-3) \cdot(-6)$. Any number $b$ divides 0 because $0=b \cdot 0$. Note carefully that ‘ $b \mid a$ ‘ is a statement about $a$ and $b$, while $c=a / b=a \cdot b^{-1}$ is a number, called the quotient of $a$ by $b$. Note also that ‘ $b$ divides $a^{\prime}$ ‘ is equivalently expressed as ‘ $b$ is a factor of $a$ ‘, and as ‘ $a$ is a multiple of $b$ ‘.
A prime number is a natural number greater than 1 which is divisible only by 1 and itself. The first few primes are ${2,3,5,7,11,13,17, \ldots}$. Eratosthenes showed how to find many primes by using his sieve. Arrange the first 1000 (or so) numbers in an orderly table, then ring the number 2 and cross off every even number; ring the next uncrossed number 3, and cross off every third number after that; ring the next uncrossed number 5 , and cross off all remaining multiples of 5 ; etc. (If your table is constructed systematically you will notice that there is a geometrical pattern in the multiples at each stage. Indeed, if you colour all multiples at any stage by using a distinctive colour, you will have created an attractive design – a visualizathe crossing off process for your table, you will be left with only the ringed prime numbers.
Finding patterns in the distribution of primes, or generating formulas for primes, has challenged people for many centuries. The ‘frequency’ of the primes falls off as we go further: to find 100 non-primes in a row, look at $101 !+2,101 !+3, \ldots, 101 !+101$. (Recall that $101 !=1 \cdot 2 \cdot 3 \cdot 4 \ldots \ldots$ $100 \cdot 101$.) But there is always a greater prime to find, as proved by Euclid. His clever and elegant idea is to take any collection of $n$ distinct primes, multiply them together and add 1 : the new number $N=\left(p_1 p_2 p_3 \cdots p_n\right)+1$ is divisible by some new prime. The ‘why’ in that proof really depends on unique factorization:

Every natural number $N>1$ can be factored uniquely (we shall state the theorem precisely later) into a product of prime numbers, and is therefore divisible by each of these primes, and by any product of a subset of them.

## 滑铁卢数学竞赛代考WATERLOO MATH CONTEST代 考|NUMBER THEORY

$i$ 使用可分性测试作为解决问题的工具。
$i i$ 使用同余符号并进行同余算术。
$i i i$ 使用费马小定理和威尔逊定理。
iv使用唯一分解定理alsoknownastheFundamentalTheoremof Arithmetic.
$v$ 使用中国剩余定理。

$A \perp B 3$
C6
D 10
$E 2$ 开甶菜问题2: 数字的最后两位是什么 $11^{111} ?$
A01
$B 11$
$C 21$
D31

## 滑铁卢数学竞赛代考WATERLOO MATH CONTEST代 考|DIVISIBILITY, PRIMES AND FACTORIZATION

Ifyourtableisconstructedsystematicallyyouwillnoticethatthereisageometricalpatterninthemultiplesateachstage. Indeed, ifyoucolourallmultiples 但正如欧几里得所证明的，总有一个更大的挈数可以找到。他聪明而优雅的想法是采取任何收藏 $n$ 不同的綁数，将它们相乘并加 1 : 新数 $N=\left(p_1 p_2 p_3 \cdots p_n\right)+1$ 能被某个新青数整除。该证明中的“为什么””实际上取决于独特的分解:

## 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。