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Number Theory

Explore the powers of divisibility, modular arithmetic, and infinity.

This course starts at the very beginning — covering all of the essential tools and concepts in number theory, and then applying them to computational art, cryptography (code-breaking), challenging logic puzzles, understanding infinity, and more!

Interactive
quizzes

46

Concepts and
exercises

410+
  1. 1

    Introduction

    In many of these warmups, if you can figure out the trick, you'll finish the problem in seconds!

    1. Last Digits

      Use shortcuts to find just the last digit of each answer – there's no need to calculate the rest!

    2. Secret Messages

      Look for patterns, and when you think you've found one, use it to decode the message!

    3. Rainbow Cycles

      Investigate the coloring rules that apply when you do math on a rainbow-striped number grid.

  2. 2

    Factorization

    Every integer greater than 1 has a unique name that can be written down in primes.

    1. Included with
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      Divisibility Shortcuts (I)

      How much can you learn about a number if you can only see its last digit?

    2. Included with
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      Divisibility Shortcuts (II)

      Review the divisibility shortcuts that apply when you're dividing by a power of 2 or 5.

    3. Included with
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      Divisibility by 9 and 3

      Explore the pattern of what remainders remain when you divide powers of 10 by 9 or 3.

    4. Included with
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      Last Digits

      Apply divisibility rules as well your own logic to determine just the last digit of each solution.

  3. 3

    GCD and LCM

    Learn how to compute and then apply your knowledge of greatest common divisors (GCDs) and least common multiples (LCMs).

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      100 Doors Revisited

      Again, imagine that long hallway of doors, but this time focus your attention on exactly who does what.

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      The LCM

      Build intuition for where least common multiples appear in both abstract and real-life contexts.

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      Billiard Tables

      Explore how the path of a ball bouncing around a pool table is affected by the table's dimensions.

    4. Included with
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      The GCD

      Use prime factorization as a tool for finding the greatest common divisors of pairs of numbers.

  4. 4

    Modular Arithmetic I

    The danger of cyclic systems: one step too far and you're back where you started!

    1. Included with
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      Times and Dates

      Time, as measured by a clock or calendar, is "modular," so let's start there...

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      Modular Congruence

      What happens when you wrap an infinite number line around a one-unit square?

    3. Included with
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      Modular Arithmetic

      Learn and practice doing arithmetic in the modular world.

    4. Included with
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      Divisibility by 11

      Review the rules for arithmetic with remainders and uncover the peculiar divisibility rule for 11.

  5. 5

    Modular Arithmetic II

    Considering the remainder "modulo" an integer is a powerful tool with many applications!

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      Additive Cycles

      Explore a concept that's lurking beneath the surface of both star drawing and decanting puzzles.

    2. Included with
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      Modular Multiplicative Inverses

      Can normal equations with no integer solutions be converted into congruences that DO have solutions?

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      Multiplicative Cycles

      What does exponentiation look like in modular arithmetic?

    4. Included with
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      Fermat's Little Theorem

      Use the factorization of a number to determine how many small numbers are relatively prime to it.

  6. 6

    Exploring Infinity

    Explore one of the most misunderstood concepts in math - infinity.

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      Counting to Infinity

      To understand cardinal infinity, first start by counting and comparing finite sets.

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      Multiple Infinities

      Explore a crazy world of numbers that contains infinitely many infinities, both small and large.

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      Hilbert's Hotel

      Help Hilbert use his hotel that has infinitely many rooms to host infinitely many sleepy guests.

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      Infinitely Large

      Can a betting game have an infinite expected value?

  7. 7

    Cryptography

    Dpef csfbljoh. → Code breaking.

    Coming Soon