Divisibility ($\mid$)

An integer $d$ divides an integer $n$, written $d\mid n$, when $n=dk$ for some integer $k$.

An integer divisor of $n$ is an integer that divides $n$.

A squarefree divisor of an integer $n$ is a divisor $d\mid n$ that is a squarefree integer. It is specified by choosing a subset of the distinct prime factors of $n$.

The greatest common divisor of integers $a$ and $b$ is the largest positive integer dividing both.

For positive integers $m,n,k$, the greatest common divisor scales by the common multiplier $k$. If $d=\gcd (m,n)$, then $kd$ is a common divisor of $km,kn$. Meanwhile Bézout's identity expresses $kd$ as an integer linear combination of $km,kn$, so their greatest common divisor divides $kd$. These two comparisons prove the identity without prime factorization.

Two integers $a$ and $b$ are coprime when $\gcd (a,b)=1$.

The least common multiple of positive integers is the smallest positive integer divisible by all of them.