Euler Phi Function of Non-Square Semiprime/Proof 1

Theorem

Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.

Let $\map \phi n$ denote the Euler $\phi$ function.

Then:

$\map \phi n = \paren {p - 1} \paren {q - 1}$

Proof

As $p$ and $q$ are distinct prime numbers, it follows that $p$ and $q$ are coprime.

Thus by Euler Phi Function is Multiplicative:

$\map \phi n = \map \phi p \, \map \phi q$

From Euler Phi Function of Prime:

$\map \phi p = p - 1$

$\map \phi q = q - 1$

Hence the result.

$\blacksquare$

Examples

Euler Phi Function of $87$

$\phi \left({87}\right) = 56$

Euler Phi Function of $91$

$\map \phi {91} = 72$

Euler Phi Function of $95$

$\phi \left({95}\right) = 72$

Euler Phi Function of $111$

$\phi \left({111}\right) = 72$

Euler Phi Function of $1257$

$\phi \left({1257}\right) = 836$