# Algebra

From The Essence Bay

### Number systems

- $$\begin{array}{l} \mathbb{N}=\{1,2,3,4,...\} \\ \mathbb{Z}=\{...,-2,-1,0,1,2,...\} \end{array}$$

### Divisibility

- If $m,n\in\mathbb{Z}$ and $m$ divides $n$ we write $m \mid n$ .

### Special types of integers

- If $n\in\mathbb{Z}$ and $2 \mid n$, then $n$ is
**even**. - If $n\in\mathbb{Z}$ and $2 \nmid n$, then $n$ is
**odd**. - If $p\in\mathbb{N}$, $p>1$ and $p$ has no positive integer divisors other than $1$ and $p$, then $p$ is
**prime**.

### Greatest common divisor

- If $a,b,c\in\mathbb{N}$ and $c \mid a,b$ then $c$ is a
**common divisor**of $a$ and $b$. - The
**greatest common divisor**of $a$ and $b$ is denoted by $gcd(a,b)$. - Two numbers $a$ and $b$ are
**coprime**if $gcd(a,b)=1$.