Euler Phi Function of 216

Example of Use of Euler $\phi$ Function

$\map \phi {216} = 72$

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:

$216 = 2^3 \times 3^3$

Thus:

\(\ds \map \phi {216}\)

\(=\)

\(\ds 216 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3}\)

\(\ds \)

\(=\)

\(\ds 216 \times \frac 1 2 \times \frac 2 3\)

\(\ds \)

\(=\)

\(\ds 36 \times 1 \times 2\)

\(\ds \)

\(=\)

\(\ds 72\)

$\blacksquare$