LIVE 33: 数论工具
7月12日起，每周二、三、四、五、一下午9:00–10:30 （您所在地当地时间）
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。
经过罗教授和课程团队反复调整，我们选择以不低于1:13的师生比进行授课。老师们将用各种方式鼓励孩子们发言、提问和思考，最大限度地提高孩子们的课堂参与度和积极性。每个班次不超过40名学生。
* 该班次直播课上课标准时间为下午5:00，纽约时间。
选择课程前，来挑战一下能力水平测试吧！
课程日历
上课日由彩色图例标识。点击任意上课日了解当天课程内容
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，学生需要在复习课程前进入Pre-Recorded课程完成周挑战测验
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
