Euler Phi Function of 1364

Example of Use of Euler $\phi$ Function

$\map \phi {1364} = 600$

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:

$1364 = 2^2 \times 11 \times 31$

Thus:

$\ds \map \phi {1364}$

$=$

$\ds 1364 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {11} } \paren {1 - \dfrac 1 {31} }$

$\ds $

$=$

$\ds 1364 \times \frac 1 2 \times \frac {10} {11} \times \frac {30} {31}$

$\ds $

$=$

$\ds 2 \times 1 \times 10 \times 30$

$\ds $

$=$

$\ds 600$

$\blacksquare$