Fungrim entry: 805c7a

\operatorname{lcm}\!\left(a, b\right) = \min\left(\left\{ m : m \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, a \mid m \,\mathbin{\operatorname{and}}\, b \mid m \right\}\right)

Assumptions:

a \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \setminus \left\{0\right\}

TeX:

\operatorname{lcm}\!\left(a, b\right) = \min\left(\left\{ m : m \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, a \mid m \,\mathbin{\operatorname{and}}\, b \mid m \right\}\right)

a \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LCM`	$\operatorname{lcm}\!\left(a, b\right)$	Least common multiple
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("805c7a"),
    Formula(Equal(LCM(a, b), Minimum(SetBuilder(m, m, And(Element(m, ZZGreaterEqual(1)), Divides(a, m), Divides(b, m)))))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(ZZ, Set(0))), Element(b, SetMinus(ZZ, Set(0))))))

Topics using this entry

Greatest common divisor