Euler Phi Function of 252

Example of Use of Euler $\phi$ Function

$\map \phi {252} = 72$

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

$\ds \map \phi n = n \prod_{p \divides n} \paren {1 - \frac 1 p}$

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

We have that:

$252 = 2^2 \times 3^2 \times 7$

Thus:

$\ds \map \phi {252}$

$=$

$\ds 252 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 7}$

$\ds $

$=$

$\ds 252 \times \frac 1 2 \times \frac 2 3 \times \frac 6 7$

$\ds $

$=$

$\ds 6 \times 1 \times 2 \times 6$

$\ds $

$=$

$\ds 72$

$\blacksquare$