Euler Phi Function of 42

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:

\(\ds \map \phi {42}\)

\(=\)

\(\ds \paren {3 - 1} \paren {7 - 1}\)

\(\ds \)

\(=\)

\(\ds 2 \times 6\)

\(\ds \)

\(=\)

\(\ds 12\)

$\blacksquare$