# Fungrim entry: 7b27cd

$\left|\left\{ k : k \in \{1, 2, \ldots n\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1 \right\}\right| = \varphi\!\left(n\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 1}$
TeX:
\left|\left\{ k : k \in \{1, 2, \ldots n\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1 \right\}\right| = \varphi\!\left(n\right)

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Cardinality$\left|S\right|$ Set cardinality
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ZZBetween$\{a, a + 1, \ldots b\}$ Integers between a and b inclusive
GCD$\gcd\!\left(n, k\right)$ Greatest common divisor
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("7b27cd"),
Formula(Equal(Cardinality(SetBuilder(k, k, And(Element(k, ZZBetween(1, n)), Equal(GCD(n, k), 1)))), Totient(n))),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(1))))

## Topics using this entry

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

2019-06-18 07:49:59.356594 UTC