Fungrim entry: d1ea57

\varphi\!\left(\operatorname{lcm}\!\left(m, n\right)\right) \varphi\!\left(\gcd\!\left(m, n\right)\right) = \varphi(m) \varphi(n)

Assumptions:

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

TeX:

\varphi\!\left(\operatorname{lcm}\!\left(m, n\right)\right) \varphi\!\left(\gcd\!\left(m, n\right)\right) = \varphi(m) \varphi(n)

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Totient`	$\varphi(n)$	Euler totient function
`LCM`	$\operatorname{lcm}\!\left(a, b\right)$	Least common multiple
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d1ea57"),
    Formula(Equal(Mul(Totient(LCM(m, n)), Totient(GCD(m, n))), Mul(Totient(m), Totient(n)))),
    Variables(m, n),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(n, ZZGreaterEqual(0)))))

Topics using this entry

Totient function