Euler Phi Function of 25,940
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Example of Use of Euler $\phi$ Function
- $\map \phi {25 \, 940} = 10 \, 368$
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:
- $25 \, 940 = 2^2 \times 5 \times 1297$
Thus:
\(\ds \map \phi {25 \, 940}\) | \(=\) | \(\ds 25 \, 940 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 5} \paren {1 - \dfrac 1 {1297} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 \, 940 \times \frac 1 2 \times \frac 4 5 \times \frac {1296} {1297}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 1 \times 4 \times 1296\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 \, 368\) |
$\blacksquare$