# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC