Let $S=\{x_1,x_2,...,x_n\}$ be some set of distinct odd integers greater than 1, such that $\forall x_i\in S$ it holds that $x_i<\sqrt{\text{lcm} (x_1,x_2,...x_n)}$ and $x_i\mid \text{lcm} (x_1,x_2,...x_n)$ (where $\mid$ means "divides").
I was wondering if:
is it possible that $1+\sum_{k=1}^{n}x_k = x_ix_j$, where $x_i\in S$ and $x_j\in S$, $x_ix_j >\sqrt{\text{lcm}\{x_1,x_2,...,x_n\}}$ and $x_ix_j\mid \text{lcm} (x_1,x_2,...x_n)$ ?
I am trying to find an example of this equality, but have not been able yet. I am wondering if this kind of equality is even possible, but I am stuck when trying to prove if such a set is possible or not. Any help on this would be really welcomed!
(It's late and it might be possible that this doesn't work. If so, let me know and I'd delete it.)
Show that this works:
$ S = \{3, 5, 9, 21, 25, 27, 35, 63\}$
$LCM = 4725, \sqrt{LCM} \approx 68.7$
$x_i = 9, x_j = 21$.