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Beginner Topic 16 of 51 Theory & Discrete Math

Number Theory Basics

Divisibility, primes and gcd.

How it works

Number theory studies integers and their properties. Core topics include divisibility, prime numbers and the fundamental theorem of arithmetic, which says every integer greater than 1 factors uniquely into primes, greatest common divisors and the Euclidean algorithm, least common multiples, congruences, Euler's totient function, Fermat's little theorem and primality testing. Many questions are easy to state yet computationally hard, such as factoring.

Step-by-step walkthrough

  1. a divides b if b = a times some integer.
  2. A prime has exactly two divisors, 1 and itself.
  3. The gcd is the largest common divisor.
  4. Euclid: gcd(48 18) = gcd(18 12) = gcd(12 6) = 6.
  5. Every number factors uniquely into primes.

Where it is used

Public-key cryptography, hashing, error-correcting codes, random number generation and algorithms in competitive programming rely on number theory.

Common pitfall

Trial division is too slow for large numbers, and naive primality tests are impractical for cryptographic sizes; use probabilistic tests such as Miller-Rabin. Overflow in big computations needs arbitrary-precision arithmetic. Confusing coprime with prime is common: 8 and 15 are coprime but not prime. Use the Euclidean algorithm for gcds instead of factoring.

Where it fits in the course

Number Theory Basics is topic 16 of 51 in Theory & Discrete Math, rated Beginner. It follows Modular Arithmetic. The next topic is Alphabets and Languages.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Theory & Discrete Math quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Number Theory Basics?
Divisibility, primes and gcd. It belongs to Theory & Discrete Math and is a beginner-level topic.
Where do people use Number Theory Basics?
Public-key cryptography, hashing, error-correcting codes, random number generation and algorithms in competitive programming rely on number theory.
What is the most common mistake with Number Theory Basics?
Trial division is too slow for large numbers, and naive primality tests are impractical for cryptographic sizes; use probabilistic tests such as Miller-Rabin.
What should I learn before this topic?
Start with Modular Arithmetic, the topic just before it in Theory & Discrete Math.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Theory & Discrete Math quiz.
Is Number Theory Basics hard to learn?
It is a Beginner topic, so it works as an entry point. If the terms are new, read the explanation twice and try the visualizer before the quiz.
How long does it take to learn Number Theory Basics?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Number Theory Basics?
Modular Arithmetic, Alphabets and Languages, Graph Coloring are closely connected. The full list is in the related topics section of this page.
What should I study after Number Theory Basics?
Continue with Alphabets and Languages, the next topic in Theory & Discrete Math.

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