Euler Phi Function of 316

Example of Use of Euler $\phi$ Function

$\map \phi {316} = 156$

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:

$316 = 2^2 \times 79$

Thus:

$\ds \map \phi {316}$

$=$

$\ds 316 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {79} }$

$\ds $

$=$

$\ds 316 \times \frac 1 2 \times \frac {78} {79}$

$\ds $

$=$

$\ds 2 \times 1 \times 78$

$\ds $

$=$

$\ds 158$

$\blacksquare$