Euler Phi Function of 164

Example of Use of Euler $\phi$ Function

$\map \phi {164} = 80$

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:

$164 = 2^2 \times 41$

Thus:

$\ds \map \phi {164}$

$=$

$\ds 164 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {41} }$

$\ds $

$=$

$\ds 164 \times \frac 1 2 \times \frac {40} {41}$

$\ds $

$=$

$\ds 2 \times 1 \times 40$

$\ds $

$=$

$\ds 80$

$\blacksquare$