Divisibility

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.

Integer factorization is the process of decomposing a composite number into a product of smaller integers, if these integers are restricted to be prime numbers then the process is called prime factorization.

This article covers factorization using trial division and fermat factorization through Pollard's Rho algorithm and using the sieve of eratosthenes.

Let $n!_{\%p}$ be a special factorial where $n!$ is divided by the maximum exponent of $p$ that divides $n!$. This article describes this problem and its solution with an implementation in C++.

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++.

Euler’s phi function represented as $\phi(n)$ gives for a number $n$ the number of coprimes in the range $[1..n]$, in other words the quantity numbers in the range $[1..n]$ whose greatest common divisor with $n$ is the unity. In this article I try to explain how it works and implement it in C++.