Euler Phi Function of 117

Example of Use of Euler $\phi$ Function

$\map \phi {117} = 72$

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:

$117 = 3^2 \times 13$

Thus:

$\ds \map \phi {117}$

$=$

$\ds 117 \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {13} }$

$\ds $

$=$

$\ds 117 \times \frac 2 3 \times \frac {12} {13}$

$\ds $

$=$

$\ds 3 \times 2 \times 12$

$\ds $

$=$

$\ds 72$

$\blacksquare$