Euler Phi Function of 76,344
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Example of Use of Euler $\phi$ Function
- $\map \phi {76 \, 338} = 25 \, 440$
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:
- $76 \, 344 = 2^3 \times 3 \times 3181$
Thus:
\(\ds \map \phi {76 \, 344}\) | \(=\) | \(\ds 76 \, 344 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {3181} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 76 \, 344 \times \frac 1 2 \times \frac 2 3 \times \frac {3180} {3181}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 1 \times 2 \times 3180\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 \, 440\) |
$\blacksquare$