Euler Phi Function of 228

Example of Use of Euler $\phi$ Function

$\map \phi {228} = 72$

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:

$228 = 2^2 \times 3 \times 19$

Thus:

$\ds \map \phi {228}$

$=$

$\ds 228 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {19} }$

$\ds $

$=$

$\ds 228 \times \frac 1 2 \times \frac 2 3 \times \frac {18} {19}$

$\ds $

$=$

$\ds 2 \times 1 \times 2 \times 18$

$\ds $

$=$

$\ds 72$

$\blacksquare$