User:Dionyziz/Sandbox/phigcd

Number theory
$$\text{Let } m = \prod p_i^{e_i^m}, n = \prod p_i^{e_i^n}, p_i \in \mathbb{P}$$

$$ \begin{array}{rcl} \phi(mn) &=& \gcd(m, n)\phi(\operatorname{lcm}(m, n)) \\ \phi(\prod p_i^{e_i^m + e_i^n}) &=& (\prod p_i^{\min(e_i^m, e_i^n)})\phi(\prod p_i^{\max(e_i^m, e_i^n)}) \\ \prod \phi(p_i^{e_i^m + e_i^n}) &=& (\prod p_i^{\min(e_i^m, e_i^n)})(\prod \phi(p_i^{\max(e_i^m, e_i^n)})) \\ \prod p_i^{e_i^m + e_i^n - 1}(p_i - 1) &=& (\prod p_i^{\min(e_i^m, e_i^n)})(\prod p_i^{\max(e_i^m, e_i^n) - 1}(p_i - 1)) \\ \prod p_i^{e_i^m + e_i^n - 1} &=& (\prod p_i^{\min(e_i^m, e_i^n)})(\prod p_i^{\max(e_i^m, e_i^n) - 1}) \\ \prod p_i^{e_i^m + e_i^n - 1} &=& \prod p_i^{\min(e_i^m, e_i^n) + \max(e_i^m, e_i^n) - 1} \\ \prod p_i^{e_i^m + e_i^n - 1} &=& \prod p_i^{e_i^m + e_i^n - 1} \\ \end{array} $$