Euler Phi Function of 84

Example of Use of Euler $\phi$ Function

$\map \phi {84} = 24$

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:

$84 = 2^2 \times 3 \times 7$

Thus:

$\ds \map \phi {84}$

$=$

$\ds 84 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 7}$

$\ds $

$=$

$\ds 84 \times \frac 1 2 \times \frac 2 3 \times \frac 6 7$

$\ds $

$=$

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

$\ds $

$=$

$\ds 24$

$\blacksquare$