Euler Phi Function of 76,326

Example of Use of Euler $\phi$ Function

$\map \phi {76 \, 326} = 25 \, 440$

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:

$76 \, 326 = 2 \times 3 \times 12 \, 721$

Thus:

$\ds \map \phi {76 \, 326}$

$=$

$\ds 76 \, 326 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {12 \, 721} }$

$\ds $

$=$

$\ds 76 \, 326 \times \frac 1 2 \times \frac 2 3 \times \frac {12 \, 720} {12 \, 721}$

$\ds $

$=$

$\ds 1 \times 2 \times 12 \, 720$

$\ds $

$=$

$\ds 25 \, 440$

$\blacksquare$