Let $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ be the prime decomposition of the integer $n$. Define $n' = n \sum_{i=1}^r \frac{\alpha_i}{p_i}$ and $\Omega(n)  = \sum_{i=1}^r \alpha_i$, $\omega(n) = r$. Let $a,b$ be relatively prime ( $(a,b) = 1$ ) and $c = a+b$. Suppose that $\Omega(c) = min(\Omega(a),\Omega(b),\Omega(c))$. Is it true, that $\Omega((a,a'))+\Omega((b,b'))+\Omega((c,c')) \le \Omega(ab)-1$? From this it would follow that
$min(\Omega(a),\Omega(b),\Omega(c)) \le \omega(abc) - 1$

Edit: From the answer given by Kevin Buzzard, one can see, that the first inequality is wrong. It is unclear to me however, if the second inequality is also wrong.