Euler Phi Function of 25,940

Example of Use of Euler $\phi$ Function

$\map \phi {25 \, 940} = 10 \, 368$

where $\phi$ denotes the Euler $\phi$ Function.

$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$

where $p \divides n$ denotes the primes which divide $n$.

We have that:

$25 \, 940 = 2^2 \times 5 \times 1297$

Thus:

$\ds \map \phi {25 \, 940}$

$=$

$\ds 25 \, 940 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 5} \paren {1 - \dfrac 1 {1297} }$

$\ds $

$=$

$\ds 25 \, 940 \times \frac 1 2 \times \frac 4 5 \times \frac {1296} {1297}$

$\ds $

$=$

$\ds 2 \times 1 \times 4 \times 1296$

$\ds $

$=$

$\ds 10 \, 368$

$\blacksquare$