Conjecture that $ \frac{\gcd(a+b,ab)}{\gcd(a,b)} \mid \gcd(a,b)$ I have discovered some exercise type conjectures which I can't prove and this is one of them:

Given positive integers $a,b$, then
  $$ \frac{\gcd(a+b,ab)}{\gcd(a,b)}\ \bigg|\ \gcd(a,b)$$

Can this be proved or disproved?

From time to time, when testing my growing math packages BigZ and Forthmath, I recognize some patterns which I can't prove or disprove (or even have the ambition to). I post them here with the hope that it will not annoy too much. I hope you can bear with it.

Taking $D=(a,b)$, then $a=DA$ and $b=DB$, $(ab,a+b)=D(A+B,ABD)$ and $(A,B)=1$. So you want to know if $(ab,a+b)/D$ divides $D$, i.e. $(A+B,ABD)|D$? well, let's see if the prime common divisors between $A+B$ and $ABD$ are divisor of $D$ as well.
If $p$ is a prime common divisor of $A+B$ and $ABD$, by Gauss lemma, $p|A$ or $p|B$ or $p|D$. If $p|A$ or $p|B$ there is a contradiction with the coprimality of $A$ and $B$ $p$ cause if $p|A$ then $p|(A+B)-A=B$. So $p|D$. 
