#### LIVE 90: Number Theory

Sa 12:00–1:30am from 9/2 (Your Time*)

This LIVE course contains a total of 30 hours of live video instruction (~$20/hour), plus 1 year of access to Prof. Loh's recorded videos from this course.

We’ve carefully designed our courses to maximize engagement. Each of our LIVE classrooms has at least one staff member for every 13 students. The typical class size is around 30 students.

* The official meeting time for this course is 8:00pm in New York.

The class time will change over the duration of this session, because your country's Daylight Savings Time adjustments differ from the target time zone of this course.

Diagnostic Tests:

### Calendar

Classes will meet on the highlighted dates. Click on any date to view the topics for that day.

Color Key:

Lesson on new topic

Discuss homework challenge

### Syllabus

The 20 course meetings are split in 16 lessons (called Day 1 through Day 16 below), and 4 homework exam discussions. Each exam discussion meeting happens after 4 lessons.

**Day 1**

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

**Day 2**

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

**Day 3**

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

**Day 4**

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

**Day 5**

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

**Day 6**

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;

**Day 7**

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

**Day 8**

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

**Day 9**

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

**Day 10**

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

**Day 11**

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

**Day 12**

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

**Day 13**

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

**Day 14**

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

**Day 15**

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

**Day 16**

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

### Co-Stars

**MICHELLE GAO**

Univ of Maryland Math Competition Honorable Mention • Plays Violin and Volleyball • Enjoys Physics and Coding • •

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**RAZZI MASROOR**

Ohio MATHCOUNTS Champion • 2022, 2023 MOP (Math Olympiad Program) Qualifier • MIT PRIMES Research Track

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