Practice
Number Theory
Courses
Take a guided, problem-solving based approach to learning Number Theory. These compilations provide unique perspectives and applications you won't find anywhere else.
Number Theory
What's inside
- Introduction
- Factorization
- GCD and LCM
- Modular Arithmetic I
- Modular Arithmetic II
- Exploring Infinity
Number Bases
What's inside
- Introduction
- Binary and Other Bases
- Digits and Divisibility
- Decimal Expansions in Base 10
- Divisibility in Other Bases
Math History
What's inside
- Introduction
- Perfecting Patterns
Additional Practice
Sharpen your skills with these quizzes designed to check your understanding of the fundamentals.
Integers
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Number Theory Warmups
If numbers aren't beautiful, we don't know what is. Dive into this fun collection to play with numbers like never before, and start unlocking the connections that are the foundation of Number Theory.
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Divisibility
You can divide 6 into equal parts of 1, 2, 3, or 6 (but not 4 or 5) because 6 is divisible by these numbers. The rules of divisibility have wide-ranging applications as an easy test for divisibility.
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Prime Numbers
2, 3, 5, 7, ... The set of prime numbers is the alphabet of mathematics that allows us to communicate across the universe.
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Prime Factorization and Divisors
Learn how to break down numbers big and small, as proposed by the Fundamental Theorem of Arithmetic.
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Greatest Common Divisor / Lowest Common Multiple
What is the largest number that can divide two numbers without a remainder? What is the smallest number that is divisible by two numbers without a remainder?
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Number Bases
How would math be different if humans only had 6 fingers and 8 toes? How does a computer do math if it has no fingers to count on?
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Factorials
2! = 2, 3! = 3*2, 4! = 4*3*2… and 100! is a lot better than writing out 158 digits. 90! is the largest factorial that can fit in a tweet.
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Integer Sequences
1, 1, 2, 3, 5, 8, 13, ... These Fibonacci numbers grow like rabbits!
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Common Misconceptions (Number Theory)
Is 0 an even number? Is 1 a prime number? Avoid these common misconceptions and learn the truth!
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Rational Numbers
Can all numbers be written as fractions? Don't be irrational - understand some of the fundamental classifications of numbers.
Modular Arithmetic
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Modular Arithmetic Operations
Considering the remainder "modulo" an integer is a powerful, foundational tool in Number Theory. You already use in clocks and work modulo 12.
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Basic Applications of Modular Arithmetic
Solve integer equations, determine remainders of powers, and much more with the power of Modular Arithmetic.
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Euler's Theorem
Euler's theorem relate to the remainder of various powers and has applications ranging from modern cryptography to recreational problem-solving.
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Arithmetic Functions
It might seem counterintuitive, but if you convolute these complicated things, they'll simplify in no time! Extend your understanding of the totient function by studying this class of functions.
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Quadratic Residues
Is there a perfect square that leaves a remainder of 7 when divided by 82? We don't need to list out all the squares, but just need to check that (7/82)=-1.
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Encryption with Number Theory
Results in number theory discovered hundreds of years ago by Fermat and Euler fuel the modern cryptography keeping your texts, emails, and other electronic data safe.
Diophantine Equations
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Linear Diophantine Equations
What amounts of money can you make with only $5 and $7 bills? Can you make $23 with those 2 types of bills?
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Quadratic Diophantine Equations
The Pythagorean Theorem is one of the most famous quadratic diophantine equations, which are equations with squared variables solved over the integers. How many solutions can you find?
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General Diophantine Equations
We have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.
Community Wiki
Browse through thousands of Number Theory wikis written by our community of experts.
Integers
- Even and Odd Numbers
- Divisibility Rules (2,3,5,7,11,13,17,19,...)
- Remainder
- Prime Numbers
- Infinitely Many Primes
- Distribution of Primes
- Mersenne Prime
- Prime Factorization
- Perfect Squares, Cubes, and Powers
- Factors
- Perfect Numbers
- Fundamental Theorem of Arithmetic
- Greatest Common Divisor
- Lowest Common Multiple
- Division Algorithm
- Euclidean Algorithm
- Bezout's Identity
- Extended Euclidean Algorithm
- Decimal Expansion
- Binary Numbers
- Hexadecimal Numbers
- Number Base
- Fractional and Non-Integer Number Bases
- Negative Integer Number Base
- Factorials
- Stirling's Formula
- Double Factorials and Multifactorials
- Trailing Number of Zeros
- Rightmost Non-zero Digit of n!
- Wilson's Theorem
- Fibonacci Sequence
- Tribonacci Sequence
- Catalan Numbers
- Is 0 even, odd, or neither?
- Is 0 a prime number?
- Is 1 prime?
- Is 2 prime?
- Modular Arithmetic Misconceptions
- Converting Decimals and Fractions
- Converting Fractions into Decimals
- Converting Repeating Decimals into Fractions
- Rational Numbers
- Irrational Numbers
- History of Irrational Numbers
- Real Numbers
- Transcendental Numbers
Modular Arithmetic
- Modular Arithmetic
- Parity of Integers
- Number Theory
- Chinese Remainder Theorem
- Fermat's Little Theorem
- Lucas' Theorem
- Sum of Squares Theorems
- Euler's Totient Function
- Euler's Theorem
- Finding the Last Digit of a Power
- Order of an Element
- Primitive Roots
- Lifting The Exponent
- Carmichael's Lambda Function
- Arithmetic Function
- Multiplicative Function
- Möbius Function
- Dirichlet Convolution
- Dirichlet Series
- Riemann Zeta Function
- Prime Zeta Function
- Cyclotomic Polynomials
- Quadratic Residues
- Legendre Symbol
- Jacobi Symbol
- Law of Quadratic Reciprocity
- Euler's Criterion
- Public-Key Cryptography
- Diffie-Hellman
- RSA Encryption
Diophantine Equations
- Linear Diophantine Equations
- System of Linear Diophantine Equations
- Postage Stamp Problem / Chicken McNugget Theorem
- Integer Equations - Stars and Bars
- Finding The Number of Digits
- Diophantine Equations - Solve by Factoring
- Diophantine Equations - Modular Arithmetic Considerations
- Fermat's Method of Infinite Descent
- Pell's Equation
- Hensel's Lemma
- Vieta Root Jumping
- General Diophantine Equations - Problem Solving