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Fungrim entry: 7b27cd

#{k:k{1,2,,n}  and  gcd ⁣(n,k)=1}=φ(n)\# \left\{ k : k \in \{1, 2, \ldots, n\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(n, k\right) = 1 \right\} = \varphi(n)
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
\# \left\{ k : k \in \{1, 2, \ldots, n\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(n, k\right) = 1 \right\} = \varphi(n)

n \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
Cardinality#S\# S Set cardinality
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Totientφ(n)\varphi(n) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Cardinality(Set(k, For(k), And(Element(k, Range(1, n)), Equal(GCD(n, k), 1)))), Totient(n))),
    Assumptions(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.

2021-03-15 19:12:00.328586 UTC