Euler Phi Function of 616

Example of Use of Euler $\phi$ Function

$\map \phi {616} = 240$

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:

$616 = 2^3 \times 7 \times 11$

Thus:

$\ds \map \phi {616}$

$=$

$\ds 616 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 7} \paren {1 - \dfrac 1 {11} }$

$\ds $

$=$

$\ds 616 \times \frac 1 2 \times \frac 6 7 \times \frac {10} {11}$

$\ds $

$=$

$\ds 4 \times 1 \times 6 \times 10$

$\ds $

$=$

$\ds 240$

$\blacksquare$