Euler Phi Function of 60

Example of Use of Euler $\phi$ Function

$\map \phi {60} = 16$

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:

$60 = 2^2 \times 3 \times 5$

Thus:

$\ds \map \phi {60}$

$=$

$\ds 60 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5}$

$\ds $

$=$

$\ds 60 \times \frac 1 2 \times \frac 2 3 \times \frac 4 5$

$\ds $

$=$

$\ds 2 \times 2 \times 4$

$\ds $

$=$

$\ds 16$

$\blacksquare$