Say we have the equations $$2b=a$$ $$3c=a$$ $$4d=a$$ $$a\neq b\neq c\neq d$$ How can these be solved to find the smallest possible integer value of $a$ with all the other unknowns as integers? Is there a general way to find this if the list of equations continues with each expression on the L.H.S. increasing by one in both coefficient and letter of the alphabet (i.e. the next equation involved is $5e=a$, the next $6f=a$, so on and so forth) while each variable remains unequal to any other?
By inspection, I noticed one possible solution is $b=12$, $c=8$, and $d=6$ for $a=24$. I then noticed these are all divisible by $2$ so a more optimal solution is $a=12$, which I think (judging by the values for the other variables) is the smallest integer value it can take given the conditions. I don't think a method of inspection would be feasible if the collection of equations extended much further however, so was wondering if there's a better method for this.
$a$ must be the least common multiple of $2$, $3$ and $4$, i.e. $12$. This is by definition the answer for this special case.