Modulo

Let $a,b \in \mathbb{Z}$, we say that $a$ divides $b$, written $a \given b$, if there’s an integer $n$ so that: $b = na$. If $a$ divides $b$ then $b$ is divisible by $a$ and $a$ is a divisor or factor of $b$, also $b$ is called a multiple of $a$.

This article covers the greatest common divisor and how to find it using the euclidean algorithm, the extended euclidean algorithm to find solutions to the equation $ax + by = gcd(a, b)$ where $a, b$ are unknowns.

Modular arithmetic is a type of arithmetic that deals with integers and remains within a fixed range of values. It involves performing arithmetic operations such as addition, subtraction, multiplication, and division, but with the added concept of a “modulus” or a “mod” value.

This article covers the definition a congruence relation, and some of its properties like addition, multiplication, exponentiation and inverse. Next I show how we can use the extended euclidean algorithm to find the modular multiplicative inverse in a general case and in the case of coprime numbers.

The extended euclidean algorithm finds solutions to the equation $ax + by = gcd(a, b)$ where $a, b$ are unknowns. This article covers a few applications of the extended euclidean algorithm like finding the modular multiplicative inverse of a number, and finding solutions for linear congruence equations.

The euclidean algorithm finds the greatest common divisor of two numbers. In this article I implement the algorithm from scratch in C++.