Euler Phi Function Of 42 - ProofWiki

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Example of Euler $\phi$ Function of Square-Free Integer

$\map \phi {42} = 12$

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

Proof

From Euler Phi Function of Square-Free Integer:

$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$

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

We have that:

$42 = 2 \times 3 \times 7$

and so is square-free.

Thus:

\(\ds \map \phi {42}\) \(=\) \(\ds \paren {3 - 1} \paren {7 - 1}\)
\(\ds \) \(=\) \(\ds 2 \times 6\)
\(\ds \) \(=\) \(\ds 12\)

$\blacksquare$

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