Euler Phi Function of 25,930

Example of Euler $\phi$ Function of Square-Free Integer

$\map \phi {25 \, 930} = 10 \, 368$

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:

$25 \, 930 = 2 \times 5 \times 2593$

and so is square-free.

Thus:

\(\ds \map \phi {25 \, 930}\)

\(=\)

\(\ds \paren {5 - 1} \paren {2593 - 1}\)

\(\ds \)

\(=\)

\(\ds 4 \times 2592\)

\(\ds \)

\(=\)

\(\ds 2^2 \times \paren {2^5 \times 3^4}\)

\(\ds \)

\(=\)

\(\ds 2^7 \times 3^4\)

\(\ds \)

\(=\)

\(\ds 10 \, 368\)

$\blacksquare$