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Fungrim entry: d1ea57

φ ⁣(lcm ⁣(m,n))φ ⁣(gcd ⁣(m,n))=φ ⁣(m)φ ⁣(n)\varphi\!\left(\operatorname{lcm}\!\left(m, n\right)\right) \varphi\!\left(\gcd\!\left(m, n\right)\right) = \varphi\!\left(m\right) \varphi\!\left(n\right)
Assumptions:mZ0andnZ0m \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\!\left(m\right) \varphi\!\left(n\right)

m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
LCMlcm ⁣(a,b)\operatorname{lcm}\!\left(a, b\right) Least common multiple
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ZZGreaterEqualZn\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.

2019-08-17 11:32:46.829430 UTC