Euler Phi Function of 824
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Example of Use of Euler $\phi$ Function
- $\map \phi {824} = 408$
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:
- $824 = 2^3 \times 103$
Thus:
\(\ds \map \phi {824}\) | \(=\) | \(\ds 824 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {103} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 824 \times \frac 1 2 \times \frac {102} {103}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 102\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 408\) |
$\blacksquare$