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Number Theory

via Brilliant


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!


  • Introduction: In many of these warmups, if you can figure out the trick, you'll finish the problem in seconds!
    • Last Digits: Use shortcuts to find just the last digit of each answer – there's no need to calculate the rest!
    • Secret Messages: Look for patterns, and when you think you've found one, use it to decode the message!
    • Rainbow Cycles: Investigate the coloring rules that apply when you do math on a rainbow-striped number grid.
  • Factorization: Every integer greater than 1 has a unique name that can be written down in primes.
    • Divisibility Shortcuts (I): How much can you learn about a number if you can only see its last digit?
    • Divisibility Shortcuts (II): Review the divisibility shortcuts that apply when you're dividing by a power of 2 or 5.
    • Divisibility by 9 and 3: Explore the pattern of what remainders remain when you divide powers of 10 by 9 or 3.
    • Last Digits: Apply divisibility rules as well your own logic to determine just the last digit of each solution.
    • Arithmetic with Remainders: How do the remainders of an operation's inputs impact the remainder of the calculation output?
    • Digital Roots: Investigate surprising patterns that surface when you calculate digital roots.
    • Factor Trees: Factor each number one step at a time until every piece that you have is a prime.
    • Prime Factorization: Learn to use factorization as a versatile problem-solving tool with primes.
    • Factoring Factorials: Since they're defined as products, what happens when you factor them?
    • Counting Divisors: Learn a quick technique for determining how many different divisors a number has.
    • 100 Doors: Imagine a long hallway with 100 closed doors numbered 1 to 100...
    • How Many Prime Numbers Are There?: Are there finitely many prime numbers or infinitely many of them, and how can you be sure either way?
  • GCD and LCM: Learn how to compute and then apply your knowledge of greatest common divisors (GCDs) and least common multiples (LCMs).
    • 100 Doors Revisited: Again, imagine that long hallway of doors, but this time focus your attention on exactly who does what.
    • The LCM: Build intuition for where least common multiples appear in both abstract and real-life contexts.
    • Billiard Tables: Explore how the path of a ball bouncing around a pool table is affected by the table's dimensions.
    • The GCD: Use prime factorization as a tool for finding the greatest common divisors of pairs of numbers.
    • Dots on the Diagonal: When you draw a right triangle on a grid of dots, how many dots does does the hypotenuse cut through?
    • Number Jumping (I): When do these jumping rules allow you to reach every number on the number line?
    • Number Jumping (II): Develop a systematic procedure to determine the smallest positive integer that you can reach by jumps.
    • Number Jumping (III): What's the pattern to these answers? What's going on in the big picture here?
    • Relating LCM and GCD: Understand how the GCD and LCM are related by thinking about factors arranged in a Venn diagram.
    • Billiard Tables Revisited (I): Explore the patterns created when pool balls "paint" the squares they touch as they roll.
    • Billiard Tables Revisited (II): How do you get back "home" to the bottom left pocket?
  • Modular Arithmetic I: The danger of cyclic systems: one step too far and you're back where you started!
    • Times and Dates: Time, as measured by a clock or calendar, is "modular," so let's start there...
    • Modular Congruence: What happens when you wrap an infinite number line around a one-unit square?
    • Modular Arithmetic: Learn and practice doing arithmetic in the modular world.
    • Divisibility by 11: Review the rules for arithmetic with remainders and uncover the peculiar divisibility rule for 11.
    • Star Drawing (I): You probably know how to draw a 5-pointed star, but what about an 8 or 12 or 30-pointed star?
    • Star Drawing (II): Learn a general formula for the number of points a star will have.
    • Star Drawing (III): How many different 60-point stars can you make, drawing just one path on a circle of 60 points?
    • Die Hard Decanting (I): The challenge is to measure out a specific quantity of liquid using only a few types of legal moves.
    • Die Hard Decanting (II): Which pairs of containers can measure out any quantity of liquid and which ones have limited use?
  • Modular Arithmetic II: Considering the remainder "modulo" an integer is a powerful tool with many applications!
    • Additive Cycles: Explore a concept that's lurking beneath the surface of both star drawing and decanting puzzles.
    • Modular Multiplicative Inverses: Can normal equations with no integer solutions be converted into congruences that DO have solutions?
    • Multiplicative Cycles: What does exponentiation look like in modular arithmetic?
    • Fermat's Little Theorem: Use the factorization of a number to determine how many small numbers are relatively prime to it.
    • Totients: This isn't Fermat's fearsome Last Theorem, but it still packs a big punch for a little guy!
    • Last Digits Revisited: Use Euler's theorem to quickly find just the last few digits of enormous exponential towers!
    • Perfect Shuffling: Leverage what you know about totients to find any card in the deck after a series of perfect shuffles.
  • Exploring Infinity: Explore one of the most misunderstood concepts in math - infinity.
    • Counting to Infinity: To understand cardinal infinity, first start by counting and comparing finite sets.
    • Multiple Infinities: Explore a crazy world of numbers that contains infinitely many infinities, both small and large.
    • Hilbert's Hotel: Help Hilbert use his hotel that has infinitely many rooms to host infinitely many sleepy guests.
    • Infinitely Large: Can a betting game have an infinite expected value?
  • Cryptography: Dpef csfbljoh. → Code breaking.


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