Euler Phi Function of 140

Example of Use of Euler $\phi$ Function

$\map \phi {140} = 48$

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:

$140 = 2^2 \times 5 \times 7$

Thus:

$\ds \map \phi {140}$

$=$

$\ds 140 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 5} \paren {1 - \dfrac 1 7}$

$\ds $

$=$

$\ds 140 \times \frac 1 2 \times \frac 4 5 \times \frac 6 7$

$\ds $

$=$

$\ds 2 \times 1 \times 4 \times 6$

$\ds $

$=$

$\ds 48$

$\blacksquare$