数论工具图标

LIVE 35: 数论工具

二、五19:00–20:30从9月13日 (美国东部时间)

报名课程 $519课程详细介绍

该模块直播课总计30小时,课时费约为(~$17/小时)。报名直播课将同时享受该模块对应录播课程(Pre-Recorded Course)一年有效,无限次回看。

课程日历

上课日期由彩色图例标识。点击任意上课日期了解当天课程内容。

图例

16节日挑战课程,每节课通过不同的数学问题学习新的知识内容

4节周挑战复习课程,集中解决周挑战测试中遇到的问题

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课程大纲

LIVE在线直播课程每个模块包含16次日挑战Session(第1课到第16课)+ 4次周挑战复习课(Weekly Challenge 1到4)共20节课程。每四节日挑战Session结束后将有一次周挑战复习课,学生需要在复习课前进入Pre-Recorded课程完成周挑战测验。

第1课

Remainder mod 10; definition of modular congruency; notation of a modulo b; remainders of n²; modular addition, subtraction and multiplication; remainders modulo 11; negative remainders

第2课

Explanation and motivation for divisibility rules for 3 and 9 and shortcuts for their use; sum and average of an arithmetic progression; triangular numbers; modular multiplication

第3课

Explanation and motivation for divisibility rules for 2, 4, and 8 and shortcuts for their use; permutations; divisibility by 12; sum and average of arithmetic progression; negative remainders

第4课

Remainders after dividing by 99; factors; patterns in multiples of 9 mod 1; palindromic numbers; factors of 1001 and 1111; negative remainders; remainders mod 11; arithmetic progression

第5课

Prime factorization; number of factors; sum of factors; average of factors; sum of reciprocals of factors; product of factors; factors of 111; expanding factors; sum of consecutive powers of 2; geometric series

第6课

Number of zeroes at the end of combinatorial expressions such as factorial; modular multiplication; floor function; ways to choose n objects; sum of consecutive powers of 2;

第7课

Least Common Multiple (LCM); Greatest Common Divisor (GCD); prime factorization; product of LCM and GCD; quotient of LCM and GCD; factorials

第8课

Motivation for and examples of Euclidean Algorithm for finding GCD; Fibonacci numbers; factors of 111; relatively prime numbers

第9课

Relatively prime numbers; pattern of cycling remainders; remainders of multiples of 2, 3, 4, 5, 6, 7, 8 and 9; Venn diagram; Inclusion / exclusion; Euler's Totient Function

第10课

Chinese Remainder Theorem and use with composite moduli; negative remainders; solving sets of congruences; LCM; remainders of multiples of 6 mod 5; negative remainders

第11课

Chinese Remainder Theorem with non-relatively-prime moduli; remainders of multiples of 9 mod 12; cycles of remainders of multiples of 9; LCM; reduction of systems of congruences; unsolvable congruences

第12课

Systems of three congruences; Euler's Totient Function; remainders modulo composite numbers; pairwise relatively prime numbers; Venn diagram; factoring; combinatorial counting

第13课

Factoring tricks for solving algebraic equations; area and perimeter of rectangles; number of ways to factor; equations in 1/x; impossibility of division by 0; number of integers solutions to an equation

第14课

Remainders of powers; cycles of remainders of powers; pattern of last two digits of powers of 7; remainders of powers of 7 mod 4; power towers

第15课

Multiplicative inverses with respect to a modulus; explanation and motivation for divisibility trick for 7; repeating cycles of remainders

第16课

Terminating decimals and their fraction representations; repeating periods of repeating decimals; proof of why square root of 2 is irrational; prime factorization; proof techiques and directions of logic; proof by contradiction

启明星助教老师

Vivian Loh 的头像
VIVIAN LOH

宾夕法尼亚州MATHCOUNTS州冠军 • 两次入围MOP(美国数学奥林匹克夏令营 ) • 2022年EGMO(欧洲女子数学奥林匹克),美国国家队选手

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Shelly Yang 的头像
SHELLY YANG

代表内华达州参加MATHCOUNTS全国赛 • 学校数学俱乐部财务主管与秘书 • 喜欢视频制作和钢琴

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