Use for questions related to gcd (greatest common divisor), lcm (least common multiple), and related concepts from integer and ring theory.

The greatest common divisor (also known as *highest common factor*) of two or more integers is the largest integer that divides all of them. It may be computed using the Euclidean algorithm.

Bézout's identity states that for non-zero integers $a$ and $b$ there exist integers $x$ and $y$ with $ax+by=\gcd(a,b)$.

If $a, b \in \mathbb{N}$, write $a \mid b$ if $a$ divides $b$, i.e. there is $k \in \mathbb{N}$ such that $b = ka$.

The least (or lowest) common multiple of $a_1, \dots, a_k \in \mathbb{N}$ is the smallest positive integer $N$ such that $a_i \mid N$ for $i = 1, \dots, k$. We usually denote $N$ by $\operatorname{lcm}(a_1, \dots, a_k)$. Note that $\operatorname{lcm}(a_1, \dots, a_k)$ can be defined recursively from a binary definition. That is,

$$\operatorname{lcm}(a_1, \dots, a_k) = \operatorname{lcm}(\operatorname{lcm}(\dots\operatorname{lcm}(\operatorname{lcm}(a_1, a_2), a_3), \dots, a_{k-1}), a_k).$$

If $a, b \in \mathbb{N}$ and $a = p_1^{r_1}\dots p_m^{r_m}$, $b = p_1^{s_1}\dots p_m^{s_n}$ are their prime decompositions (where some of the $r_i$ and $s_j$ can be zero), we have

$$\operatorname{lcm}(a, b) = p_1^{\max(r_1, s_1)}\dots\ p_m^{\max(r_m, s_m)}.$$

Note that $\operatorname{lcm}(a, b)\operatorname{gcd}(a, b) = ab$.

All of these notions can be generalised to any commutative ring; the above is just the particular case of (positive elements of) the ring $\mathbb{Z}$.