Euler Phi Function of 666 equals Product of Digits
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Theorem
The number $666$ has the following interesting property:
- $\map \phi {666} = 6 \times 6 \times 6$
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:
- $666 = 2 \times 3^2 \times 37$
Thus:
\(\ds \map \phi {666}\) | \(=\) | \(\ds 666 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {37} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 666 \times \frac 1 2 \times \frac 2 3 \times \frac {36} {37}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 1 \times 2 \times 36\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 2 \times \paren {2^2 \times 3^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 216\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 \times 6 \times 6\) |
$\blacksquare$
Also see
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $666$