Euler Phi Function of 264

Example of Use of Euler $\phi$ Function

$\map \phi {264} = 80$

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:

$264 = 2^3 \times 3 \times 11$

Thus:

$\ds \map \phi {264}$

$=$

$\ds 264 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {11} }$

$\ds $

$=$

$\ds 264 \times \frac 1 2 \times \frac 2 3 \times \frac {10} {11}$

$\ds $

$=$

$\ds 4 \times 1 \times 2 \times 10$

$\ds $

$=$

$\ds 80$

$\blacksquare$