Euler Phi Function of 524

Example of Use of Euler $\phi$ Function

$\map \phi {524} = 260$

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

Proof

From Euler Phi Function of Integer:

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

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

We have that:

$524 = 2^2 \times 131$

Thus:

\(\ds \map \phi {524}\)

\(=\)

\(\ds 524 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {131} }\)

\(\ds \)

\(=\)

\(\ds 524 \times \frac 1 2 \times \frac {130} {131}\)

\(\ds \)

\(=\)

\(\ds 2 \times 1 \times 130\)

\(\ds \)

\(=\)

\(\ds 260\)

$\blacksquare$