Euler Phi Function of 2025

Example of Use of Euler $\phi$ Function

$\map \phi {2025} = 1080$

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:

$2025 = 3^4 \times 5^2$

Thus:

\(\ds \map \phi {2025}\) \(=\) \(\ds 2025 \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 5}\)
\(\ds \) \(=\) \(\ds 2025 \times \dfrac 2 3 \times \dfrac 4 5\)
\(\ds \) \(=\) \(\ds 1080\)

$\blacksquare$

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