Study Notes

Divisibility rules:
- A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 5 if its last digit is 0 or 5.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
Prime numbers:
- A prime number is a positive integer greater than 1 that is divisible only by 1 and itself.
- Examples: 2, 3, 5, 7, 11, ...
- Prime numbers play a crucial role in number theory, as they serve as the building blocks for all other numbers.

GCD definition:
- The GCD of two or more integers is the largest integer that divides each of them without leaving a remainder.
GCD calculation methods:
- Euclidean algorithm: a method for finding the GCD of two numbers by repeatedly replacing (a, b) with (b, a mod b) until b = 0, then taking a as the GCD.
- Prime factorization: find the GCD by finding the common prime factors of the numbers.

Congruence definition:
- Two integers a and b are congruent modulo n (denoted by a ≡ b (mod n)) if they have the same remainder when divided by n.
Diophantine equation definition:
- A Diophantine equation is an equation in two or more variables of the form ax + by = c, where a, b, and c are integers.
- Examples: 2x + 3y = 7, 5x - 2y = 11

Fermat's Little Theorem:
- If p is a prime number, then a^p ≡ a (mod p) for any integer a not divisible by p.
Euler's Theorem:
- If a and n are coprime (have no common factors), then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.

Diophantine approximation:
- The study of approximating irrational numbers using rational numbers.
Continued fractions:
- A way of representing a number as a sequence of integers, often used in Diophantine approximation.
- Example: the continued fraction representation of the golden ratio φ is [1; 1, 1, 1, ...]

나눗셈의 법칙:
- 마지막 자리의 숫자가 0, 2, 4, 6, 또는 8이라면 2로 나눌 수 있습니다.
- 자릿수의 합이 3으로 나눌 수 있다면 3으로 나눌 수 있습니다.
- 마지막 자리의 숫자가 0 또는 5라면 5로 나눌 수 있습니다.
- 자릿수의 합이 9으로 나눌 수 있다면 9로 나눌 수 있습니다.
소수:
- 1보다 큰 자연수에서 1과 자기 자신만으로 나눌 수 있는 수입니다.
- 예: 2, 3, 5, 7, 11, ...
- 소수는 모든 수의 기본構成要素입니다.

최대공약수 정의:
- 두 개 이상의 정수의 최대공약수는 각수를 나누어 나머지가 없는 최대 정수입니다.
최대공약수 계산 방법:
- 오일러 알고리즘: 두 수의 최대공약수를 구하는 알고리즘입니다. (a, b) -> (b, a mod b) -> ... -> (a, 0), b=a
- 소인수분해: 공통의 소인수를 찾아 최대공약수를 구합니다.