Euler Phi Function of 26

Example of Use of Euler $\phi$ Function

$\map \phi {26} = 12$

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:

$26 = 2 \times 13$

Thus:

\(\ds \map \phi {26}\)

\(=\)

\(\ds 26 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {13} }\)

\(\ds \)

\(=\)

\(\ds 26 \times \dfrac 1 2 \times \dfrac {12} {13}\)

\(\ds \)

\(=\)

\(\ds 12\)

$\blacksquare$