Euler Phi Function of 666 equals Product of Digits

Theorem

The number $666$ has the following interesting property:

$\map \phi {666} = 6 \times 6 \times 6$

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

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

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

We have that:

$666 = 2 \times 3^2 \times 37$

Thus:

$\ds \map \phi {666}$

$=$

$\ds 666 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {37} }$

$\ds $

$=$

$\ds 666 \times \frac 1 2 \times \frac 2 3 \times \frac {36} {37}$

$\ds $

$=$

$\ds 3 \times 1 \times 2 \times 36$

$\ds $

$=$

$\ds 3 \times 2 \times \paren {2^2 \times 3^2}$

$\ds $

$=$

$\ds 216$

$\ds $

$=$

$\ds 6 \times 6 \times 6$

$\blacksquare$