Euler Phi Function of 5188

Example of Use of Euler $\phi$ Function

$\map \phi {5188} = 2592$

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:

$5188 = 2^2 \times 1297$

Thus:

$\ds \map \phi {5188}$

$=$

$\ds 5188 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {1297} }$

$\ds $

$=$

$\ds 5188 \times \frac 1 2 \times \frac {1296} {1297}$

$\ds $

$=$

$\ds 2 \times 1 \times 1296$

$\ds $

$=$

$\ds 2592$

$\blacksquare$