Fungrim entry: fbe121

\operatorname{lcm}\!\left(r s, c\right) = \frac{\operatorname{lcm}\!\left(r, c\right) \operatorname{lcm}\!\left(s, c\right)}{\left|c\right|}

Assumptions:

r \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, s \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \gcd\!\left(r, s\right) = 1 \,\mathbin{\operatorname{and}}\, c \ne 0

TeX:

\operatorname{lcm}\!\left(r s, c\right) = \frac{\operatorname{lcm}\!\left(r, c\right) \operatorname{lcm}\!\left(s, c\right)}{\left|c\right|}

r \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, s \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \gcd\!\left(r, s\right) = 1 \,\mathbin{\operatorname{and}}\, c \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`LCM`	$\operatorname{lcm}\!\left(a, b\right)$	Least common multiple
`Abs`	$\left\|z\right\|$	Absolute value
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(n, k\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("fbe121"),
    Formula(Equal(LCM(Mul(r, s), c), Div(Mul(LCM(r, c), LCM(s, c)), Abs(c)))),
    Variables(r, s, c),
    Assumptions(And(Element(r, ZZ), Element(s, ZZ), Element(c, ZZ), Equal(GCD(r, s), 1), Unequal(c, 0))))

Topics using this entry

Greatest common divisor