5 of 5 attempt to add some context

Minimize $a+b+c+d$ given $\text{LCM}(a,b,c,d)=1000$

It is the inverse of this problem.

Minimize $a+b+c+d$ given $a, b, c, d$ are distinct positive integers, and $\operatorname{lcm}(a,b,c,d) =1000$.

Of course, it would be possible to enumerate all quadruplets with $\operatorname{lcm}(a,b,c,d)=1000$, however it might be interesting to see whether there is a more intelligent approach. Or even if the minimum is found by an exhaustive search, at least the bound can possibly be proved more directly. (As in the answer to the post which inspired this: Minimize $LCM(a,b,c,d)$ given $a+b+c+d=1000$.)

Since $1000=2^3\cdot5^3$, we will need at least one of the numbers $a$, $b$, $c$, $d$ to contain factor $2^3$. And, similarly, at least one of them has to contain $5^3$.