 Solution

Use prime factorization and multiplicativity to check the identity one prime at a time. Fix $p$ and put $a=v_p(m)$, $b=v_p(n)$, $r=\min (a,b)$. The local factors to compare areIf either exponent is zero, $r=0$ and the equality follows from $\varphi (1)=1$. If both are positive, the prime-power formula makes both sidesMultiplying the local identities over all primes and using $\gcd (m,n)={\prod }_p{p}^{\min (v_p(m),v_p(n))}$ givesThis accounts for repeated prime factors shared by the two inputs, which ordinary coprime multiplicativity alone would not remove.