Euler Phi Function of 418
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Example of Euler $\phi$ Function of Square-Free Integer
- $\map \phi {418} = 180$
where $\phi$ denotes the Euler $\phi$ Function.
Proof
From Euler Phi Function of Square-Free Integer:
- $\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
- $418 = 2 \times 11 \times 19$
and so is square-free.
Thus:
\(\ds \map \phi {418}\) | \(=\) | \(\ds \paren {11 - 1} \paren {19 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 \times 18\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 5} \times \paren {2 \times 3^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3^2 \times 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 180\) |
$\blacksquare$