Euler Phi Function/Examples/20

Example of Use of Euler $\phi$ Function

$\map \phi {20} = 8$

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:

$20 = 2^2 \times 5$

Thus:

\(\ds \map \phi {20}\)

\(=\)

\(\ds 20 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 5}\)

\(\ds \)

\(=\)

\(\ds 20 \times \dfrac 1 2 \times \dfrac 4 5\)

\(\ds \)

\(=\)

\(\ds 8\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): phi function (totient function)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's phi function (phi function, totient function)