Totient identity for a product and a greatest common divisor
= Totient identity for a product and a greatest common divisor
{title2=$\phi(mn)\phi((m,n))=\phi(m)\phi(n)(m,n)$}
The prime-power formula and <multiplicativity> of <Euler's totient function> imply $\phi(mn)\phi(\gcd(m,n))=\phi(m)\phi(n)\gcd(m,n)$. 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 $\phi(1)=1$.