#### LIVE 35: Number Theory

Tu/F 7:00–8:30pm from 9/13 (USA Eastern Time)

This LIVE course contains a total of 30 hours of live video instruction (~$17/hour), plus 1 year of access to Prof. Loh's recorded videos from this course.

### 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 exam assigned for homework

### 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

**VIVIAN LOH**

Pennsylvania MATHCOUNTS Champion • 2-time MOP qualifier • Member of US team at 2022 EGMO

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**SHELLY YANG**

National MATHCOUNTS competitor for Nevada • Trecretary of school math club • Enjoys video production and piano

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