What is the smallest possible value of the least common multiple of $a, b, c, d$ if we know that these four numbers are distinct and $a + b + c + d = 1000$?
Problem
Source: 2019 Dürer Math Competition Finals Day2 E11 E+10 https://artofproblemsolving.com/community/c1621835_2019_
Tags: least common multiple, LCM, number theory