Euler Phi Function of 825

Example of Use of Euler $\phi$ Function

$\map \phi {825} = 400$

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:

$825 = 3 \times 5^2 \times 11$

Thus:

$\ds \map \phi {825}$

$=$

$\ds 825 \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5} \paren {1 - \dfrac 1 {11} }$

$\ds $

$=$

$\ds 825 \times \frac 2 3 \times \frac 4 5 \times \frac {10} {11}$

$\ds $

$=$

$\ds 5 \times 2 \times 4 \times 10$

$\ds $

$=$

$\ds 5 \times 2 \times 2^2 \times \paren {2 \times 5}$

$\ds $

$=$

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

$\ds $

$=$

$\ds \paren {2^2 \times 5}^2$

$\ds $

$=$

$\ds 400$

$\blacksquare$