Euler Phi Function of 25,942
Jump to navigation
Jump to search
Example of Euler $\phi$ Function of Square-Free Integer
- $\map \phi {25 \, 942} = 10 \, 368$
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:
- $25 \, 942 = 2 \times 7 \times 17 \times 109$
and so is square-free.
Thus:
\(\ds \map \phi {25 \, 942}\) | \(=\) | \(\ds \paren {7 - 1} \paren {17 - 1} \paren {109 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 \times 16 \times 108\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 3} \times 2^4 \times \paren {2^2 \times 3^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^7 \times 3^4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 \, 368\) |
$\blacksquare$