Euler Phi Function Of 135 - ProofWiki

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Example of Use of Euler $\phi$ Function

$\map \phi {135} = 72$

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:

$135 = 3^3 \times 5$

Thus:

\(\ds \map \phi {135}\) \(=\) \(\ds 135 \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5}\)
\(\ds \) \(=\) \(\ds 135 \times \frac 2 3 \times \frac 4 5\)
\(\ds \) \(=\) \(\ds 9 \times 2 \times 4\)
\(\ds \) \(=\) \(\ds 72\)

$\blacksquare$

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