Euler Phi Function Of 90 - ProofWiki

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Example of Use of Euler $\phi$ Function

$\map \phi {90} = 24$

where $\phi$ denotes the Euler $\phi$ Function.

From Euler Phi Function of Integer:

$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$

where $p \divides n$ denotes the primes which divide $n$.

We have that:

$90 = 2 \times 3^2 \times 5$

Thus:

$\ds \map \phi {90}$	$=$	$\ds 90 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5}$
$\ds $	$=$	$\ds 90 \times \frac 1 2 \times \frac 2 3 \times \frac 4 5$
$\ds $	$=$	$\ds 3 \times 1 \times 2 \times 4$
$\ds $	$=$	$\ds 24$

$\blacksquare$

Từ khóa » Phi 90

\(\ds \map \phi {90}\)	\(=\)	\(\ds 90 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5}\)
\(\ds \)	\(=\)	\(\ds 90 \times \frac 1 2 \times \frac 2 3 \times \frac 4 5\)
\(\ds \)	\(=\)	\(\ds 3 \times 1 \times 2 \times 4\)
\(\ds \)	\(=\)	\(\ds 24\)