Totient identity for a product and a greatest common divisor ($\varphi (mn)\varphi ((m,n))=\varphi (m)\varphi (n)(m,n)$)

The prime-power formula and multiplicativity of Euler's totient function imply $\varphi (mn)\varphi (\gcd (m,n))=\varphi (m)\varphi (n)\gcd (m,n)$. For a prime with positive exponents $a,b$ in the inputs, both local products equal $p^{a+b+\min (a,b)-2}(p-1{)}^2$. If one exponent is zero, the equality is immediate from $\varphi (1)=1$.