Euler Phi Function of 945

Example of Use of Euler $\phi$ Function

$\map \phi {945} = 432$

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:

$945 = 3^3 \times 5 \times 7$

Thus:

$\ds \map \phi {945}$

$=$

$\ds 945 \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5} \paren {1 - \dfrac 1 7}$

$\ds $

$=$

$\ds 945 \times \frac 2 3 \times \frac 4 5 \times \frac 6 7$

$\ds $

$=$

$\ds 9 \times 2 \times 4 \times 6$

$\ds $

$=$

$\ds 432$

$\blacksquare$