Number Theory

Our most challenging course. A systematic introduction to abstract concepts across number theory, illustrated through custom-designed problems that make difficult-to-grasp ideas concrete. Covers concepts in MATHCOUNTS, AMC 8, AMC 10, AMC 12, and AIME.

If you haven't yet seen our revolutionary teaching method, visit our home page first, to shatter your expectations of what online math looks like.

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Who Is This Course For?

Are you taking any math class from 6th Grade Math through Calculus, and already know the content in our Combinatorics class? If so, then you are already ready to understand this course. It will efficiently teach you the challenging concepts in Number Theory which are not taught in regular school curriculum.

Diagnostic Tests:

Will I understand?Will I learn new things?

Course Structure

This course contains 20 LIVE 1-hour video sessions, of which 16 are lessons, and 4 are exam review sessions that discuss the course's 4 practice 1-hour exams (homework). Each class is co-taught by 2 of our Stars. In total, there are over 150 problems.

Each lesson contains several problems which are meant to be unfamiliar to you, so that they inspire you to invent your own solutions. In our LIVE video class format, you propose your ideas, and we brainstorm together with you to construct a variety of alternative solutions using the class's own ideas. This actively involves you in discovering mathematics.

Each practice exam contains 20 problems, of which half are of average difficulty, and the other half are very challenging.

Reviews

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

AH

Brilliant-Wonderful, thought-out well explanations to questions I never thought could be so interesting! I've learned to love to find out more with this wonderful course.

JS

Thank you so much for creating this course. I love it, and it's the best. If you don't believe me, here's a piece of evidence. On the final test of this course, I scored a 100%. This proves that this course is very effective and helps you with number theory! - Jocelyn!

AZ

Worth everything and learned a lot

Syllabus

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

Enroll

We have LIVE and self-paced versions of this course. We strongly recommend the LIVE format, as it has been carefully designed to be so engaging that students fall in love with challenge. There has never been anything like it before. All times are in the USA Eastern Time Zone, and will follow the USA Daylight Savings Time adjustment on Sun Nov 6.

enroll

For students who cannot match our schedule, or who are already particularly motivated, our self-paced version contains all of Prof. Loh's teaching on this subject. Enroll by choosing the format you prefer:

Self-Paced