Euler Phi Function/Examples/Phi is 6

Example of Use of Euler $\phi$ Function

There are $4$ numbers for which the value of the Euler $\phi$ function is $6$:

$7, 9, 14, 18$

Proof

From Euler Phi Function of $9$, we already have that $\map \phi 9 = 6$.

$7$

From Euler Phi Function of Prime:

$\map \phi 7 = 7 - 1 = 6$

$14$

From Euler Phi Function of Integer:

\(\ds \map \phi {14}\)

\(=\)

\(\ds 14 \paren {1 - \dfrac 1 7} \paren {1 - \dfrac 1 2}\)

\(\ds \)

\(=\)

\(\ds 14 \times \dfrac 6 7 \times \dfrac 1 2\)

\(\ds \)

\(=\)

\(\ds 6\)

$18$

From Euler Phi Function of Integer:

\(\ds \map \phi {18}\)

\(=\)

\(\ds 18 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3}\)

\(\ds \)

\(=\)

\(\ds 18 \times \dfrac 1 2 \times \dfrac 2 3\)

\(\ds \)

\(=\)

\(\ds 6\)

Hence the result.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $18$