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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!

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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

#### Divisibility Shortcuts (I)

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

2. Included with

#### Divisibility Shortcuts (II)

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

3. Included with

#### 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

#### 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).

1. Included with

#### 100 Doors Revisited

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

2. Included with

#### The LCM

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

3. Included with

#### Billiard Tables

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

4. Included with

#### 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

#### Times and Dates

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

2. Included with

#### Modular Congruence

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

3. Included with

#### Modular Arithmetic

Learn and practice doing arithmetic in the modular world.

4. Included with

#### 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!

1. Included with

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

2. Included with

#### Modular Multiplicative Inverses

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

3. Included with

#### Multiplicative Cycles

What does exponentiation look like in modular arithmetic?

4. Included with

#### 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.

1. Included with

#### Counting to Infinity

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

2. Included with

#### Multiple Infinities

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

3. Included with

#### Hilbert's Hotel

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

4. Included with