Euler Phi Function/Examples/24

Example of Use of Euler $\phi$ Function

$\map \phi {24} = 8$

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:

$24 = 2^3 \times 3$

Thus:

\(\ds \map \phi {24}\)

\(=\)

\(\ds 24 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3}\)

\(\ds \)

\(=\)

\(\ds 24 \times \dfrac 1 2 \times \dfrac 2 3\)

\(\ds \)

\(=\)

\(\ds 8\)

$\blacksquare$