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Fungrim entry: db4763

(gcd⁡ ⁣(m,n)=1)    ⟹    (φ ⁣(mn)=φ(m)φ(n))\left(\gcd\!\left(m, n\right) = 1\right) \;\implies\; \left(\varphi\!\left(m n\right) = \varphi(m) \varphi(n)\right)
Assumptions:m∈Z≥0  and⁡  n∈Z≥0m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
TeX:
\left(\gcd\!\left(m, n\right) = 1\right) \;\implies\; \left(\varphi\!\left(m n\right) = \varphi(m) \varphi(n)\right)

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
GCDgcd⁡ ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Totientφ(n)\varphi(n) Euler totient function
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("db4763"),
    Formula(Implies(Equal(GCD(m, n), 1), Equal(Totient(Mul(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