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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(m) \varphi(n)
Assumptions:mZ0  and  nZ0m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
\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}
Fungrim symbol Notation Short description
Totientφ(n)\varphi(n) 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:
    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)))))

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2021-03-15 19:12:00.328586 UTC