Fungrim entry: 805c7a

\operatorname{lcm}\!\left(a, b\right) = \min \left\{ m : m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; a \mid m \;\mathbin{\operatorname{and}}\; b \mid m \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\{ m : m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; a \mid m \;\mathbin{\operatorname{and}}\; b \mid m \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`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`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(Set(m, For(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