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

Greatest common divisor