$a+b+c+ab+bc+ca+abc=1000$ find the minimum value of $a+b+c$ Given $a+b+c+ab+bc+ca+abc=1000$.
Find the minimum value of $a+b+c$.
Now we are considering $a$, $b$, $c$ to be integers and here in lies the pertinent problem as I could not get an answer in integers but in fractions.

I am assuming here $a=b=c$ since $1000$ is the cube of $10$.
$$(a+1)(b+1)(c+1) = 1000$$
now, $(a+1)^3 = 1000$;
$$a+1 = 10\iff
a=9$$
so, $a+b+c = 3*9 = 27$. 
Just that we can get the integer value but I cannot say $27$ is the minimum value for $a+b+c$.
