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Fungrim entry: 56d7fe

φ ⁣(mn)=φ ⁣(m)φ ⁣(n)gcd ⁣(m,n)φ ⁣(gcd ⁣(m,n))\varphi\!\left(m n\right) = \frac{\varphi\!\left(m\right) \varphi\!\left(n\right) \gcd\!\left(m, n\right)}{\varphi\!\left(\gcd\!\left(m, n\right)\right)}
Assumptions:mZ1andnZ1m \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}
TeX:
\varphi\!\left(m n\right) = \frac{\varphi\!\left(m\right) \varphi\!\left(n\right) \gcd\!\left(m, n\right)}{\varphi\!\left(\gcd\!\left(m, n\right)\right)}

m \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
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("56d7fe"),
    Formula(Equal(Totient(Mul(m, n)), Div(Mul(Mul(Totient(m), Totient(n)), GCD(m, n)), Totient(GCD(m, n))))),
    Variables(m, n),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(1)))))

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

2019-08-19 14:38:23.809000 UTC