[Thgx0312555]

$\Rightarrow $
สมมติ $3 \nmid n$
และ $ n = p_1^ {i_1}p_2^{i_2}...p_k^{i_k} $ไม่มี $p_i $ใดๆ $= 3$

$\phi (n) = n(1-\dfrac{1}{p_1})(1-\dfrac{1}{p_2})...(1-\dfrac{1}{p_k})$
$\phi (3n) = 3n\dfrac{2}{3}(1-\dfrac{1}{p_1})(1-\dfrac{1}{p_2})...(1-\dfrac{1}{p_k})$

$\phi (3n) = 2n(1-\dfrac{1}{p_1})(1-\dfrac{1}{p_2})...(1-\dfrac{1}{p_k})$

$\therefore \phi (3n) \not= 3 \phi(n)$

$\Leftarrow $

สมมติ $3|n$
และ$ n = 3^{i_0}p_1^ {i_1}p_2^{i_2}...p_k^{i_k} $ไม่มี $p_i $ใดๆ $= 3$

$\phi (n) = n\dfrac{2}{3}(1-\dfrac{1}{p_1})(1-\dfrac{1}{p_2})...(1-\dfrac{1}{p_k})$
$\phi (3n) = 3n\dfrac{2}{3}(1-\dfrac{1}{p_1})(1-\dfrac{1}{p_2})...(1-\dfrac{1}{p_k})$

$\therefore \phi (3n) = 3 \phi(n)$

$\therefore \phi (3n) = 3 \phi(n) \Leftrightarrow 3|n$