Euler Phi Function of 72

Example of Use of Euler $\phi$ Function

$\map \phi {72} = 24$

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:

$72 = 2^3 \times 3^2$

Thus:

\(\ds \map \phi {72}\)

\(=\)

\(\ds 72 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3}\)

\(\ds \)

\(=\)

\(\ds 72 \times \frac 1 2 \times \frac 2 3\)

\(\ds \)

\(=\)

\(\ds 12 \times 1 \times 2\)

\(\ds \)

\(=\)

\(\ds 24\)

$\blacksquare$