Euler Phi Function of 5187

Example of Euler $\phi$ Function of Square-Free Integer

$\map \phi {5187} = 2592$

where $\phi$ denotes the Euler $\phi$ Function.

$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$

where $p \divides n$ denotes the primes which divide $n$.

We have that:

$5187 = 3 \times 7 \times 13 \times 19$

Thus:

$\ds \map \phi {5187}$

$=$

$\ds \paren {3 - 1} \paren {7 - 1} \paren {13 - 1} \paren {19 - 1}$

$\ds $

$=$

$\ds 2 \times 6 \times 12 \times 18$

$\ds $

$=$

$\ds 2 \times \paren {2 \times 3} \times \paren {2^2 \times 3} \times \paren {2 \times 3^2}$

$\ds $

$=$

$\ds 2^5 \times 3^4$

$\ds $

$=$

$\ds 2592$

$\blacksquare$