Euler Phi Function of 76,326
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Example of Use of Euler $\phi$ Function
- $\map \phi {76 \, 326} = 25 \, 440$
where $\phi$ denotes the Euler $\phi$ Function.
Proof
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:
- $76 \, 326 = 2 \times 3 \times 12 \, 721$
Thus:
\(\ds \map \phi {76 \, 326}\) | \(=\) | \(\ds 76 \, 326 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {12 \, 721} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 76 \, 326 \times \frac 1 2 \times \frac 2 3 \times \frac {12 \, 720} {12 \, 721}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times 2 \times 12 \, 720\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 \, 440\) |
$\blacksquare$